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

<b>let </b><font color="Maroon" title="c1">U1</font>, <font color="Maroon" title="c2">U2</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font>  st <font color="Olive" title="b1">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> holds <br/>(  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Olive" title="b1">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font> &amp;  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Olive" title="b1">f</font> <a href="alg_1.html#R2" title="ALG_1:pred.2">is_monomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font> )</span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> implies (  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font> &amp;  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R2" title="ALG_1:pred.2">is_monomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font> ) )</span><br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font> &amp;  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R2" title="ALG_1:pred.2">is_monomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font> )</span><br/>

<b>set </b><font color="Maroon" title="c4">qa</font> =  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">cqa</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">u1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">F</font> =  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font>;<br/>
<a NAME="E2:37"/><i><font color="Green" title="E32">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U2</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E3:37"/><i><font color="Green" title="E33">A3</font></i>: 
 <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R2" title="ALG_1:pred.2">is_monomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<a NAME="E1:37_1"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 
<b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="alg_1.html#D1" target="_self" title="ALG_1:def.1">Def1</a></i>;<br/>
<a NAME="E2:37_1"/>
 <a href="alg_1.html#K6" title="ALG_1:func.6">Nat_Hom</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>, <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="alg_1.html#T21" target="_self" title="ALG_1:th.21">Th21</a></i>;<br/>
<a NAME="E3:37_1"/><b>then </b>
<font color="Maroon" title="c1">U1</font>, <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 
<b>by </b><i><a class="ref" href="alg_1.html#D1" target="_self" title="ALG_1:def.1">Def1</a></i>;<br/>
<a NAME="E4:37_1"/><b>then </b><i><font color="Green" title="E35">A5</font></i>: 
 <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2" title="UNIALG_2:def.2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E5:37_1"/><b>then </b>
 <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c2">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:37_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="unialg_2.html#D2" title="UNIALG_2:def.2">UNIALG_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:37_1"/>
 <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 <b>by </b><i><a class="ref" href="unialg_2.html#D2" title="UNIALG_2:def.2">UNIALG_2:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="alg_1.html#D1" target="_self" title="ALG_1:def.1">ALG_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> holds <br/>for <font color="Olive" title="b2">o1</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b3">o2</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U2</font>  st <font color="Olive" title="b2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>for <font color="Olive" title="b4">x</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">o1</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b4">x</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> implies for <font color="Olive" title="b1">o1</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">o2</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U2</font>  st <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Olive" title="b2">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> holds <br/>for <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">o1</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b3">x</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E7:37_1"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o1</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">o2</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U2</font>  st <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Olive" title="b2">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> holds <br/>for <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">o1</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b3">x</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c9">oq</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o2</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U2</font>  st <font color="Maroon" title="c9">oq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Olive" title="b1">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> holds <br/>for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">oq</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">oq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c10">o2</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">oq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c10">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> implies for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">oq</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">oq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E8:37_1"/><i><font color="Green" title="E37">A7</font></i>: 
( <font color="Maroon" title="c9">oq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c10">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">oq</font> holds <br/><span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">oq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c11">x</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">oq</font> implies <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">oq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E9:37_1"/><i><font color="Green" title="E38">A8</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">oq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">oq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c11">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c12">x1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E12:37_1"/><i><font color="Green" title="E39">A9</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p3">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p3">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">o1</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U1</font> <b>by </b><i><a class="txt" href="#E4:37_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="#E7:37_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">y</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">U1</font><b> such that </b><br/><a NAME="E15:37_1"/><i><font color="Green" title="E40">A10</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="alg_1.html#R6" title="ALG_1:pred.6">is_representatives_FS</a> <font color="Maroon" title="c13">x1</font>
 <b>by </b><i><a class="ref" href="alg_1.html#T19" target="_self" title="ALG_1:th.19">Th19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c15">y</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E17:37_1"/><i><font color="Green" title="E41">A11</font></i>: 
<font color="Maroon" title="c9">oq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="alg_1.html#K3" title="ALG_1:func.3">QuotOp</a> <font color="Maroon" title="c14">o1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:37_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="#E8:37_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="alg_1.html#D13" target="_self" title="ALG_1:def.13">Def13</a></i>;<br/>
<b>then </b> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">oq</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="alg_1.html#D12" target="_self" title="ALG_1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o1</font> </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></i>
;<br/>
<a NAME="E19:37_1"/><b>then </b>
 ex <font color="Olive" title="b1">w</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o1</font> )
 
<b>by </b><i><a class="txt" href="#E9:37_1"><i><font color="Green" title="E38">A8</font></i></a></i>;<br/>
<a NAME="E20:37_1"/><b>then </b><i><font color="Green" title="E42">A12</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">y</font> )
 
<b>by </b><i><a class="txt" href="#E15:37_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="alg_1.html#D11" target="_self" title="ALG_1:def.11">Def11</a></i>;<br/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">o1</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T2" title="UNIALG_2:th.2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c14">o1</font> </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></i>
;<br/>
<a NAME="E22:37_1"/><b>then </b><i><font color="Green" title="E43">A13</font></i>: 
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">o1</font>
 
<b>by </b><i><a class="txt" href="#E20:37_1"><i><font color="Green" title="E42">A12</font></i></a></i>;<br/>
<a NAME="E23:37_1"/><b>then </b>
( <font color="Maroon" title="c14">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">o1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">o1y</font> = <font color="Maroon" title="c14">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> ;<br/>
<i><font color="Green" title="E44">A14</font></i>: <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">oq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c17">o1y</font></span>)</span>
<b>by </b><i><a class="txt" href="#E9:37_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="#E15:37_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="#E17:37_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="alg_1.html#D12" target="_self" title="ALG_1:def.12">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">y</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="alg_1.html#D17" target="_self" title="ALG_1:def.17">Def17</a></i>
;<br/>
<a NAME="E26:37_1"/><i><font color="Green" title="E45">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">y</font>
 
<b>by </b><i><a class="txt" href="#E20:37_1"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="alg_1.html#T1" target="_self" title="ALG_1:th.1">Th1</a></i>;<br/>
<a NAME="E27:37_1"/><i><font color="Green" title="E46">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="alg_1.html#T1" target="_self" title="ALG_1:th.1">Th1</a></i>;<br/>
<a NAME="E28:37_1"/><i><font color="Green" title="E47">X</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="#E26:37_1"><i><font color="Green" title="E45">A15</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">y</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> )</span><br/><b>assume </b><a NAME="E1:37_1_4"/><i><font color="Green" title="E48">A17</font></i>: 
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font></span><br/><a NAME="E2:37_1_4"/><b>then </b><i><font color="Green" title="E49">A18</font></i>: 
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">y</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:37_1_4"/><i><font color="Green" title="E50">A19</font></i>: 
( <font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span> &amp; <font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E26:37_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="#E27:37_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="#E1:37_1_4"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:37_1_4"/><i><font color="Green" title="E51">A20</font></i>: 
<font color="Maroon" title="c13">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E15:37_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="#E2:37_1_4"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="alg_1.html#D11" target="_self" title="ALG_1:def.11">Def11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">ym</font> = <font color="Maroon" title="c16">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <b>by </b><i><a class="txt" href="#E2:37_1_4"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">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" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> = 
<span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c19">ym</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:37_1_4"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="#E4:37_1_4"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="alg_1.html#T1" target="_self" title="ALG_1:th.1">Th1</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="alg_1.html#D17" target="_self" title="ALG_1:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font>
<b>by </b><i><a class="txt" href="#E3:37_1_4"><i><font color="Green" title="E50">A19</font></i></a>, <a class="ref" href="alg_1.html#T1" target="_self" title="ALG_1:th.1">Th1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E30:37_1"/><b>then </b>
<font color="Maroon" title="c10">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c16">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E26:37_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="#E27:37_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a>, <a class="txt" href="#E28:37_1"><i><font color="Green" title="E47">X</font></i></a></i>;<br/>
<b>hence </b><a NAME="E31:37_1"/>
<span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">oq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E4:37_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="#E7:37_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="#E8:37_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="#E12:37_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="#E22:37_1"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E25:37_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="alg_1.html#D1" target="_self" title="ALG_1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<a NAME="E4:37"/>
 <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> )</span><br/>



<b>assume </b><a NAME="E1:37_2"/><i><font color="Green" title="E34">A21</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">y1</font> = <font color="Maroon" title="c9">y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <b>by </b><i><a class="txt" href="#E2:37"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:37_2"/><i><font color="Green" title="E35">A22</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> &amp; <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c12">a</font> )
 <b>by </b><i><a class="txt" href="#E1:37_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c12">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E35">A22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:37_2"/><i><font color="Green" title="E36">A23</font></i>: 
( <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> &amp; <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c14">b</font> )
 <b>by </b><i><a class="txt" href="#E1:37_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c14">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <b>by </b><i><a class="txt" href="#E7:37_2"><i><font color="Green" title="E36">A23</font></i></a></i>;<br/>
<a NAME="E9:37_2"/>
( <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font> &amp; <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">b</font> )
 
<b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E4:37_2"><i><font color="Green" title="E35">A22</font></i></a>, <a class="txt" href="#E7:37_2"><i><font color="Green" title="E36">A23</font></i></a>, <a class="ref" href="alg_1.html#D17" target="_self" title="ALG_1:def.17">Def17</a></i>;<br/>
<a NAME="E10:37_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E1:37_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="alg_1.html#D16" target="_self" title="ALG_1:def.16">Def16</a></i>;<br/>
<b>hence </b><a NAME="E11:37_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E35">A22</font></i></a>, <a class="txt" href="#E7:37_2"><i><font color="Green" title="E36">A23</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:37"/>
 <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R2" title="ALG_1:pred.2">is_monomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font>
 <b>by </b><i><a class="txt" href="#E3:37"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="alg_1.html#D2" target="_self" title="ALG_1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
