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

<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">u2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U2</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">a</font> being   <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>  st $1 <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="Olive" title="b1">a</font> holds <br/>$2 <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="Olive" title="b1">a</font>;<br/>
<a NAME="E1:37_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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 <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/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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="c2">U2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><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> ;<br/>

<b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c8">x</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 <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>then reconsider </b><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c8">x</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> ;<br/>
<b>consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:37_1_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c10">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="c9">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="c10">a</font> )
 <b>by </b><i><a class="txt" href="alg_1.html#E1:37_1_1_1"><i><font color="Green" title="E31">A3</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="c11">a</font> = <font color="Maroon" title="c10">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="alg_1.html#E4:37_1_1_1"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font>;<br/>

<b>thus </b><a NAME="E6:37_1_1_1"/>
<font color="Maroon" title="c12">y</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="c2">U2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c13">b</font> be    <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/>

<b>assume </b><a NAME="E7:37_1_1_1"/>
<font color="Maroon" title="c8">x</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="c13">b</font>
 ;<br/>

<a NAME="E8:37_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c11">a</font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E4:37_1_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/>
<a NAME="E9:37_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c11">a</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="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<b>hence </b><a NAME="E10:37_1_1_1"/>
<font color="Maroon" title="c12">y</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">b</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E1:37"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="alg_1.html#D16" target="_self" title="ALG_1:def.16">Def16</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:37_1_1"/><i><font color="Green" title="E31">A5</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">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 <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> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">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; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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 <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/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="alg_1.html#E1:37_1_1"><i><font color="Green" title="E30">A2</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">F</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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>,<font color="Maroon" title="c2">U2</font> <b>by </b><i><a class="txt" href="alg_1.html#E3:37_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">F</font>
;<br/>

<b>let </b><font color="Maroon" title="c10">a</font> be    <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/>

<a NAME="E5:37_1_1"/>
 <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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:37_1_1"/>
<font color="Maroon" title="c9">F</font> <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="c10">a</font></span>)</span> <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="c10">a</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E3:37_1_1"><i><font color="Green" title="E31">A5</font></i></a></i>;<br/>


</div>
