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

<b>let </b><font color="Maroon">U1</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">U2</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 ;<br/>

<b>set </b><font color="Maroon">qa</font> =  <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">cqa</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">u1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font>;<br/>
<a NAME="E2:38"/><i><font color="Green">E34</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E3:38"/><i><font color="Green">E35</font></i>: 
 <a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a>  <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">U2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<a NAME="E1:38_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:38_1"/>
 <a href="alg_1.html#K6" target="_self">Nat_Hom</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span> <a href="alg_1.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>, <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:38_1"/><b>then </b>
<font color="Maroon">U1</font>, <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:38_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E5:38_1"/><b>then </b>
 <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="alg_1.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:38_1"/>
 <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a> 
 <b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="alg_1.html#D1" target="_self">ALG_1:def 1</a><br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E7:38_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">oq</font> be   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font>;<br/>



<b>assume </b><a NAME="E8:38_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">oq</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E9:38_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">oq</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E12:38_1"/><i><font color="Green">E50</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p2">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p3">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p3">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o1</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font> <b>by </b><i><a class="txt" href="alg_1.html#E3:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="alg_1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">U1</font><b> such that </b><br/><a NAME="E15:38_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> <a href="alg_1.html#R6" target="_self">is_representatives_FS</a> <font color="Maroon">x1</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E17:38_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">oq</font> <a href="hidden.html#R1">=</a>  <a href="alg_1.html#K3" target="_self">QuotOp</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="alg_1.html#T2" target="_self">Th2</a>, , </i>;<br/>
<b>then </b> <a href="relset_1.html#K4">dom</a> <font color="Maroon">oq</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>
;<br/>
<a NAME="E19:38_1"/><b>then </b>
 ex <font color="Olive">w</font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  st <br/>( <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E20:38_1"/><b>then </b><i><font color="Green">E71</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="alg_1.html#T4" target="_self">Th4</a>, </i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>
;<br/>
<a NAME="E22:38_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E23:38_1"/><b>then </b>
( <font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">o1y</font> = <font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> ;<br/>
<i><font color="Green">E73</font></i>: <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">oq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">o1y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="alg_1.html#T4" target="_self">Th4</a>, <a class="txt" href="alg_1.html#E1:38_1"><i><font color="Green">E41</font></i></a>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E26:38_1"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E27:38_1"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:38_1_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span>
 ;<br/><a NAME="E2:38_1_4"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:38_1_4"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="alg_1.html#E4:38_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="alg_1.html#E7:38_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:38_1_4"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="alg_1.html#T4" target="_self">Th4</a>, <a class="txt" href="alg_1.html#E8:38_1"><i><font color="Green">E48</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">ym</font> = <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <b>by </b><i><a class="txt" href="alg_1.html#E8:38_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">ym</font></span>)</span>
<b>by </b><i><a class="txt" href="alg_1.html#E9:38_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="alg_1.html#E12:38_1"><i><font color="Green">E50</font></i></a>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="alg_1.html#E9:38_1"><i><font color="Green">E49</font></i></a>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E29:38_1"/><b>then </b>
<font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="alg_1.html#E4:38_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<b>hence </b><a NAME="E30:38_1"/>
<span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">oq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="alg_1.html#E3:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="alg_1.html#T2" target="_self">Th2</a>, , , <a class="ref" href="alg_1.html#T5" target="_self">Th5</a>, , </i>;<br/>


</div><b>end;</b></div>

<a NAME="E4:38"/>
 <a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E87</font></i>: 
( <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="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:38_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="alg_1.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <b>by </b><i><a class="txt" href="alg_1.html#E3:38"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:38_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="alg_1.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <b>by </b><i><a class="ref" href="alg_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:38_2"/>
( <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="txt" href="alg_1.html#E3:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="alg_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E10:38_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="txt" href="alg_1.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="alg_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E11:38_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E3:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="alg_1.html#T2" target="_self">Th2</a>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:38"/>
 <a href="alg_1.html#K8" target="_self">HomQuot</a> <font color="Maroon">f</font> <a href="alg_1.html#R2" target="_self">is_monomorphism</a>  <a href="alg_1.html#K5" target="_self">QuotUnivAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" target="_self">Cng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">U2</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E1:38"><i><font color="Green">E33</font></i></a>, </i>;<br/>


</div>
