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

<b>let </b><font color="Maroon">I</font> be  <a href="group_1.html#V7">commutative</a> non <a href="vectsp_1.html#V10">degenerated</a> <a href="vectsp_2.html#V2">domRing-like</a> <a href="funcsdom.html#NM1">Ring</a>;<br/>

<a NAME="E1:103"/><i><font color="Green">E20</font></i>: 
 <a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font> is <a href="quofield.html#V3" target="_self">RingMonomorphism</a>
 
<b>by </b><i><a class="ref" href="quofield.html#D10" target="_self">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">F'</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">f'</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">I</font>,<font color="Maroon">F'</font>;<br/><b>assume </b><a NAME="E1:103_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f'</font> is <a href="quofield.html#V3" target="_self">RingMonomorphism</a>
 ;<br/><a NAME="E2:103_1"/><b>then </b>
<font color="Maroon">f'</font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 <b>by </b><i><a class="txt" href="quofield.html#E11:103_1"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E3:103_1"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">f'</font> is <a href="grcat_1.html#V1">additive</a> &amp; <font color="Maroon">f'</font> is <a href="group_6.html#V1">multiplicative</a> &amp; <font color="Maroon">f'</font> is <a href="endalg.html#V1">unity-preserving</a> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">hh</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> </span>}</span> ;<br/><a NAME="E4:103_1"/><i><font color="Green">E25</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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="103_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:103_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"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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="103_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="E1:103_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"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> </span>}</span> 
 ;<br/>

<a NAME="E2:103_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"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 
;<br/>
<b>hence </b><a NAME="E3:103_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="E2:103_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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="E5:103_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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="103_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:103_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"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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="103_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="E1:103_1_2_1"/><i><font color="Green">E34</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"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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="E3:103_1_2_1"/><i><font color="Green">E39</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"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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="E5:103_1_2_1"/><i><font color="Green">E40</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"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E41</font></i>: <font color="Maroon">a</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </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="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<i><font color="Green">E53</font></i>: <font color="Maroon">a</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y2</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </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="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<i><font color="Green">E54</font></i>: <font color="Maroon">x1</font> = 
<font color="Maroon">a</font> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <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="quofield.html#T18" target="_self">Th18</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<font color="Maroon">x2</font> = 
<font color="Maroon">a</font> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>then </b><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#T19" target="_self">Th19</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="E11:103_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="E2:103_1_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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">hh</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</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#T17" target="_self">Th17</a></i>;<br/><a NAME="E7:103_1"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">hh</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_3"/><i><font color="Green">E56</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">hh</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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="E1:103_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">hh</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="E3:103_1_3_1"/><i><font color="Green">E57</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">hh</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="E5:103_1_3_1"/><i><font color="Green">E58</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"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>
<font color="Maroon">x</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </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="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E7:103_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:103_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>  <a href="quofield.html#K1" target="_self">Q.</a> <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">hh</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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="E1:103_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>
 ;<br/>

<b>then 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="E3:103_1_3_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:103_1_3_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">hh</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:103_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">hh</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="E3:103_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">hh</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:103_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">hh</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:103_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">hh</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 <font color="Maroon">F'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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="E1:103_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">hh</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="E3:103_1_4_1"/><i><font color="Green">E60</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">hh</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="E5:103_1_4_1"/><i><font color="Green">E61</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"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>
<font color="Maroon">y</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </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="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E7:103_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 <font color="Maroon">F'</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:103_1_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">hh</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F'</font>
 <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">hh</font> = <font color="Maroon">hh</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F'</font> <b>by </b><i>, <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="E10:103_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> holds  <font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>;<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="E2:103_1_5"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>
<a NAME="E3:103_1_5"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E4:103_1_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">hh</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:103_1_5"/>
<font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">h</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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="E11:103_1"/><i><font color="Green">E66</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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="103_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_6"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</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"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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">v</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="103_1_6_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:103_1_6_1"/>
<font color="Maroon">v</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"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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="E2:103_1_6_1"/><b>then </b>
 ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> st <br/>( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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="E3:103_1_6_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">v</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="E2:103_1_6"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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="E12:103_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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="103_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_7"/>
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"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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="103_1_7_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="E1:103_1_7_1"/><i><font color="Green">E67</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"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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">u1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:103_1_7_1"/><i><font color="Green">E71</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"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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">u2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:103_1_7_1"/><i><font color="Green">E72</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"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E73</font></i>: <font color="Maroon">a</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u1</font></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#T18" target="_self">Th18</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<font color="Maroon">a</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u2</font></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#T19" target="_self">Th19</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E8:103_1_7_1"/><b>then </b>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u2</font>
 
<b>by </b><i><a class="ref" href="quofield.html#T20" target="_self">Th20</a>, </i>;<br/>
<a NAME="E9:103_1_7_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:103_1_7_1"/>
<font color="Maroon">f'</font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 
<b>by </b><i><a class="ref" href="quofield.html#T15" target="_self">Th15</a>, <a class="txt" href="quofield.html#E11:103_1"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E11:103_1_7_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">f'</font> is <a href="grcat_1.html#V1">additive</a> &amp; <font color="Maroon">f'</font> is <a href="group_6.html#V1">multiplicative</a> &amp; <font color="Maroon">f'</font> is <a href="endalg.html#V1">unity-preserving</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E12:103_1_7_1"/>
( <font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> &amp; <font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E13:103_1_7_1"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F'</font> )
 
<b>by </b><i><a class="ref" href="quofield.html#T15" target="_self">Th15</a>, <a class="ref" href="quofield.html#T11" target="_self">Th11</a></i>;<br/>
<i><font color="Green">E180</font></i>: <font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="quofield.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T21" target="_self">Th21</a>, <a class="ref" href="quofield.html#D12" target="_self">Def12</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#D12" target="_self">Def12</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="quofield.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u2</font>
<b>by </b><i/>
;<br/>
<font color="Maroon">b1</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u1</font></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#T18" target="_self">Th18</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u2</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<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#T19" target="_self">Th19</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="E16:103_1_7_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="E2:103_1_7"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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">h</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> :  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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/>;<br/><a NAME="E14:103_1"/><i><font color="Green">E181</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_8"/><i><font color="Green">E182</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">h</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_8_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="E1:103_1_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</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="E3:103_1_8_1"/><i><font color="Green">E183</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">h</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">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:103_1_8_1"/><i><font color="Green">E184</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"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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#T19" target="_self">Th19</a></i>;<br/>
<font color="Maroon">x</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></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#T20" target="_self">Th20</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</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="E7:103_1_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:103_1_8"/>
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="quofield.html#K9" target="_self">Quot.</a> <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">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_8_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="E1:103_1_8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:103_1_8_2"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:103_1_8_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font>
 
;<br/>
<b>hence </b><a NAME="E5:103_1_8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="quofield.html#T19" target="_self">Th19</a>, <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="E3:103_1_8"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:103_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_9"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_9"/>
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">h</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 <font color="Maroon">F'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_9_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="E1:103_1_9_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</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="E3:103_1_9_1"/><i><font color="Green">E186</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">h</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">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:103_1_9_1"/><i><font color="Green">E187</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"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</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#T18" target="_self">Th18</a></i>;<br/>
<font color="Maroon">y</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></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#T19" target="_self">Th19</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</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="E7:103_1_9_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F'</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:103_1_9"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F'</font>
 <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">h</font> = <font color="Maroon">h</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F'</font> <b>by </b><i>, <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/><b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM6">Function</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>,<font color="Maroon">F'</font> ;<br/><a NAME="E18:103_1"/><i><font color="Green">E188</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</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/> for <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Olive">u</font> holds <br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_10"><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 <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/><b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:103_1_10"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font>
 ;<br/>

<a NAME="E2:103_1_10"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font>
 
;<br/>
<b>hence </b><a NAME="E3:103_1_10"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="quofield.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:103_1"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">h</font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_11"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_11"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</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> holds <br/> ( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K10">*</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_11_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 <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>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</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/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font> ;<br/>
<b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:103_1_11_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">v</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:103_1_11_1"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</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/>
<a NAME="E7:103_1_11_1"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> &amp; <font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:103_1_11_1"/><b>then </b><i><font color="Green">E194</font></i>: 
( <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F'</font> )
 
<b>by </b><i><a class="ref" href="quofield.html#T15" target="_self">Th15</a>, <a class="ref" href="quofield.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E9:103_1_11_1"/><i><font color="Green">E195</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="quofield.html#D12" target="_self">Def12</a>, <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<i><font color="Green">E196</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a>, <a class="ref" href="quofield.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a>, <a class="ref" href="quofield.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></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">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="quofield.html#E10:103_1"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
<b>by </b><i/>
;<br/>
<a NAME="E12:103_1_11_1"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K17" target="_self">quotadd</a> <font color="Maroon">I</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font> ;<br/>
<i><font color="Green">E198</font></i>: <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K10" target="_self">qadd</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font></span>)</span> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p2">(<span class="default"><a href="quofield.html#K6" target="_self">padd</a> <font color="Maroon">u</font>,<font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T20" target="_self">Th20</a>, <a class="ref" href="quofield.html#T21" target="_self">Th21</a>, <a class="ref" href="quofield.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font></span>)</span>

;<br/>
<b>reconsider </b><font color="Maroon">t'</font> =  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</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/>
<a NAME="E16:103_1_11_1"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> <b>by </b><i>, </i>;<br/>
<i><font color="Green">E201</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a>, <a class="ref" href="quofield.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a>, <a class="ref" href="quofield.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="quofield.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="quofield.html#K22" target="_self">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
<b>by </b><i/>
;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font> ;<br/>
<i><font color="Green">E202</font></i>: <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K11" target="_self">qmult</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font></span>)</span> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p2">(<span class="default"><a href="quofield.html#K7" target="_self">pmult</a> <font color="Maroon">u</font>,<font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T20" target="_self">Th20</a>, <a class="ref" href="quofield.html#T21" target="_self">Th21</a>, </i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font></span>)</span>

;<br/>
<b>reconsider </b><font color="Maroon">t'</font> =  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</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/>
<a NAME="E22:103_1_11_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E23:103_1_11_1"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K14" target="_self">q1.</a> <font color="Maroon">I</font></span>)</span>
 
<b>by </b><i><a class="txt" href="quofield.html#E7:103_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E24:103_1_11_1"/><i><font color="Green">E205</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<a NAME="E25:103_1_11_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<a NAME="E27:103_1_11_1"/><i><font color="Green">E206</font></i>: 
 <a href="quofield.html#K14" target="_self">q1.</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_11_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_11_1_5"/><i><font color="Green">E207</font></i>: 
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>  <a href="quofield.html#K14" target="_self">q1.</a> <font color="Maroon">I</font> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_11_1_5_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="E1:103_1_11_1_5_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K14" target="_self">q1.</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> ;<br/>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> 
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> 
<b>by </b><i><a class="ref" href="quofield.html#T30" target="_self">Th30</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:103_1_11_1_5_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:103_1_11_1_5"/>
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>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K14" target="_self">q1.</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_11_1_5_2"><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="E1:103_1_11_1_5_2"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> ;<br/>
<font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T30" target="_self">Th30</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> 
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E4:103_1_11_1_5_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K14" target="_self">q1.</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:103_1_11_1_5"/>
 <a href="quofield.html#K14" target="_self">q1.</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="quofield.html#T29" target="_self">Th29</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">t'</font> =  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</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/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span></span>)</span> = 
<font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font>
<b>by </b><i><a class="ref" href="quofield.html#T26" target="_self">Th26</a>, <a class="ref" href="quofield.html#T28" target="_self">Th28</a>, <a class="ref" href="quofield.html#T18" target="_self">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="endalg.html#D8">ENDALG:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="endalg.html#D8">ENDALG:def 8</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font>
<b>by </b><i><a class="ref" href="quofield.html#T27" target="_self">Th27</a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
;<br/>
<b>hence </b><a NAME="E30:103_1_11_1"/>
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font> )
 <b>by </b><i><a class="ref" href="quofield.html#D13" target="_self">Def13</a>, <a class="ref" href="quofield.html#D14" target="_self">Def14</a>, <a class="ref" href="quofield.html#D15" target="_self">Def15</a>, <a class="ref" href="quofield.html#T22" target="_self">Th22</a>, <a class="ref" href="quofield.html#T23" target="_self">Th23</a>, <a class="ref" href="quofield.html#T24" target="_self">Th24</a>, <a class="ref" href="quofield.html#T25" target="_self">Th25</a>, <a class="ref" href="quofield.html#T4" target="_self">Th4</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:103_1_11"/><b>then </b>
( <font color="Maroon">h</font> is <a href="grcat_1.html#V1">additive</a> &amp; <font color="Maroon">h</font> is <a href="group_6.html#V1">multiplicative</a> &amp; <font color="Maroon">h</font> is <a href="endalg.html#V1">unity-preserving</a> )
 
<b>by </b><i><a class="ref" href="endalg.html#D8">ENDALG:def 8</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a>, <a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:103_1_11"/>
<font color="Maroon">h</font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E20:103_1"/><i><font color="Green">E210</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_12"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_12"/><i><font color="Green">E211</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<a NAME="E2:103_1_12"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:103_1_12"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font>
 <b>by </b><i><a class="ref" href="quofield.html#T21" target="_self">Th21</a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:103_1"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_13"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_13"/><i><font color="Green">E213</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/>
<a NAME="E2:103_1_13"/><i><font color="Green">E214</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="relset_1.html#K4">dom</a> <font color="Maroon">f'</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_13_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="E1:103_1_13_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f'</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="E3:103_1_13_1"/><i><font color="Green">E215</font></i>: 
<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/>
<a NAME="E4:103_1_13_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:103_1_13_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 
;<br/>
<b>hence </b><a NAME="E6:103_1_13_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:103_1_13"/><i><font color="Green">E216</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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_13_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="E1:103_1_13_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</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="E3:103_1_13_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>hence </b><a NAME="E4:103_1_13_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:103_1_13"/>
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="relset_1.html#K4">dom</a> <font color="Maroon">f'</font> holds <br/><font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_13_3"><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="E1:103_1_13_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f'</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/>
<b>reconsider </b><font color="Maroon">u</font> = <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="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="quofield.html#D12" target="_self">Def12</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">u'</font> =  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</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/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p2">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font>
<b>by </b><i><a class="ref" href="quofield.html#D12" target="_self">Def12</a>, </i>
<br/>.= 
<font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F'</font></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="quofield.html#T20" target="_self">Th20</a>, <a class="ref" href="endalg.html#D8">ENDALG:def 8</a></i>
<br/>.= 
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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="E6:103_1_13_3"/>
<font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:103_1_13"/>
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f'</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h'</font> be   <a href="struct_0.html#NM6">Function</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>,<font color="Maroon">F'</font>;<br/><b>assume </b><a NAME="E1:103_1_14"/><i><font color="Green">E218</font></i>: 
( <font color="Maroon">h'</font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Maroon">h'</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f'</font> )
 ;<br/><a NAME="E2:103_1_14"/><i><font color="Green">E219</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/><a NAME="E3:103_1_14"/>
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 <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> holds <br/><font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_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="E1:103_1_14_1"/>
<font color="Maroon">x</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/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</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/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font> ;<br/>
<b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:103_1_14_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">u</font>
 <b>by </b><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="E7:103_1_14_1"/><i><font color="Green">E221</font></i>: 
( <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="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E222</font></i>: <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">hh</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="quofield.html#T18" target="_self">Th18</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="quofield.html#D12" target="_self">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h'</font> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#D12" target="_self">Def12</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">a'</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="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">b'</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">a''</font> =  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">a'</font>, <font color="Maroon">b''</font> =  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">b'</font> as   <a href="struct_0.html#NM1">Element</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/>
<a NAME="E11:103_1_14_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">b''</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <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="103_1_14_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:103_1_14_1_2"/>
<font color="Maroon">b''</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <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/>

<a NAME="E2:103_1_14_1_2"/><b>then </b><i><font color="Green">E228</font></i>: 
<font color="Maroon">b''</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K13" target="_self">q0.</a> <font color="Maroon">I</font>
 
;<br/>
<a NAME="E3:103_1_14_1_2"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b''</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">b</font> = 
<font color="Maroon">b'</font> <a href="quofield.html#K2" target="_self">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="quofield.html#T22" target="_self">Th22</a>, <a class="ref" href="quofield.html#T23" target="_self">Th23</a>, </i>
;<br/>
<b>hence </b><a NAME="E5:103_1_14_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="quofield.html#D13" target="_self">Def13</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">aa</font> =  <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>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM1">Element</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/>
<b>reconsider </b><font color="Maroon">bb</font> =  <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">b</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM1">Element</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/>
<i><font color="Green">E232</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#D13" target="_self">Def13</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">aa</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a''</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a''</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#D13" target="_self">Def13</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a''</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">bb</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a''</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b''</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a''</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">b''</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#D12" target="_self">Def12</a>, <a class="ref" href="quofield.html#D15" target="_self">Def15</a>, </i>
;<br/>
<b>reconsider </b><font color="Maroon">bi</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="quofield.html#D13" target="_self">Def13</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">bi'</font> =  <a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">bi</font> as   <a href="struct_0.html#NM1">Element</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/>
<a NAME="E17:103_1_14_1"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a''</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">b''</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K18" target="_self">quotmult</a> <font color="Maroon">I</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a''</font>,<font color="Maroon">bi'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="quofield.html#D13" target="_self">Def13</a>, <a class="ref" href="quofield.html#D15" target="_self">Def15</a>, </i>;<br/>
<i><font color="Green">E236</font></i>: <font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K18" target="_self">quotmult</a> <font color="Maroon">I</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a''</font>,<font color="Maroon">bi'</font></span>)</span> = 
<font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K11" target="_self">qmult</a> <span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">a'</font></span>)</span>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <font color="Maroon">bi</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p2">(<span class="default"><a href="quofield.html#K7" target="_self">pmult</a> <font color="Maroon">a'</font>,<font color="Maroon">bi</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E19:103_1_14_1"/>
 <a href="quofield.html#K7" target="_self">pmult</a> <font color="Maroon">a'</font>,<font color="Maroon">bi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_14_1_5"><b>proof </b></a><div class="add">

 <a href="quofield.html#K7" target="_self">pmult</a> <font color="Maroon">a'</font>,<font color="Maroon">bi</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">bi</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">bi</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a'</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">bi</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">bi</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>
<b>hence </b><a NAME="E2:103_1_14_1_5"/>
 <a href="quofield.html#K7" target="_self">pmult</a> <font color="Maroon">a'</font>,<font color="Maroon">bi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="quofield.html#D13" target="_self">Def13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:103_1_14_1"/>
<font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="quofield.html#D14" target="_self">Def14</a>, <a class="ref" href="quofield.html#T22" target="_self">Th22</a>, <a class="ref" href="quofield.html#T23" target="_self">Th23</a>, <a class="ref" href="quofield.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:103_1_14"/>
<font color="Maroon">h'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E23:103_1"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</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>,<font color="Maroon">F'</font> st <br/>( <font color="Olive">h</font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Olive">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f'</font> &amp; ( for <font color="Olive">h'</font> being  <a href="struct_0.html#NM6">Function</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>,<font color="Maroon">F'</font>  st <font color="Olive">h'</font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Olive">h'</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">f'</font> holds <br/><font color="Olive">h'</font> <a href="funct_2.html#R2">=</a> <font color="Olive">h</font> ) )
 <b>by </b><i><a class="ref" href="quofield.html#T19" target="_self">Th19</a>, <a class="ref" href="quofield.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:103"/><b>then </b>
<font color="Maroon">I</font> <a href="quofield.html#R3" target="_self">has_Field_of_Quotients_Pair</a>  <a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font>, <a href="quofield.html#K24" target="_self">canHom</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="quofield.html#T14" target="_self">Th14</a>, <a class="ref" href="quofield.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E4:103"/>
 ex <font color="Olive">F</font> being non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a>  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">I</font>,<font color="Olive">F</font> st <font color="Maroon">I</font> <a href="quofield.html#R3" target="_self">has_Field_of_Quotients_Pair</a> <font color="Olive">F</font>,<font color="Olive">f</font>
 ;<br/>


</div>
