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

<b>set </b><font color="Maroon">f</font> =  <a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</font>;<br/>
<a NAME="E1:42_1_1"/>
 <a href="pralg_1.html#K12">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="waybel26.html#T31">WAYBEL26:31</a></i>;<br/>
<b>then </b>the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel18.html#D3">WAYBEL18:def 3</a></i>
<br/>.= 
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="card_3.html#T20">CARD_3:20</a></i>
;<br/>
<a NAME="E3:42_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel26.html#T33">WAYBEL26:33</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:42_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:42_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> )
 ;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">x1</font>, <font color="Maroon">a2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">a1</font>, <font color="Maroon">f2</font> = <font color="Maroon">a2</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T2">WAYBEL26:2</a></i>;<br/><b>assume </b><a NAME="E5:42_1_1_2"/>
<span class="p1">(<span class="default"><a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/><b>then </b> <a href="funct_6.html#K10">commute</a> <font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i/>
<br/>.= 
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">f2</font>
<b>by </b><i/>
;<br/><a NAME="E7:42_1_1_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/><b>hence </b><a NAME="E8:42_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:42_1_1"/>
 <a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>

<a NAME="E6:42_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</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"><span class="p2">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">M</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_1_1_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">M</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:42_1_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">M</font></span>)</span> ;<br/>
<a NAME="E4:42_1_1_3"/><i><font color="Green">E50</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel26.html#T33">WAYBEL26:33</a>, <a class="ref" href="yellow_1.html#T28">YELLOW_1:28</a></i>;<br/>
<a NAME="E5:42_1_1_3"/><b>then </b><i><font color="Green">E53</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T63">FUNCT_5:63</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> =  <a href="funct_6.html#K10">commute</a> <font color="Maroon">x</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T34">WAYBEL26:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T2">WAYBEL26:2</a></i>;<br/>
<a NAME="E9:42_1_1_3"/>
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/>
<a NAME="E10:42_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</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="waybel26.html#K1">oContMaps</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="waybel_5.html#K7">=&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:42_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:42_1_1"/>
 <a href="waybel29.html#K5" target="_self">commute</a> <font color="Maroon">X</font>,<font color="Maroon">M</font>,<font color="Maroon">Y</font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>


</div>
