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

<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font>;<br/>
<a NAME="E1:42_1_1"/>
 <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="ref" href="waybel26.html#T31" title="WAYBEL26:th.31">WAYBEL26:31</a></i>;<br/>
<b>then </b>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel18.html#D3" title="WAYBEL18:def.3">WAYBEL18:def 3</a></i>
<br/>.= 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font>
<b>by </b><i><a class="ref" href="card_3.html#T20" title="CARD_3:th.20">CARD_3:20</a></i>
;<br/>
<a NAME="E3:42_1_1"/><b>then </b><i><font color="Green" title="E20">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel26.html#T33" title="WAYBEL26:th.33">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" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.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">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</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="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">x2</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="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font></span><br/><b>let </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</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"> ( <font color="Maroon" title="c5">x1</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="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">x2</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="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> implies <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> )</span><br/><b>assume </b><a NAME="E2:42_1_1_2"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c5">x1</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="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">x2</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="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></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"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> implies <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> )</span><br/><b>then reconsider </b><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c8">a2</font> = <font color="Maroon" title="c6">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c10">f2</font> = <font color="Maroon" title="c8">a2</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T2" title="WAYBEL26:th.2">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" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font></span><br/><b>then </b> <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c9">f1</font> = 
<span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
<b>by </b><i><a class="ref" href="waybel29.html#D5" target="_self" title="WAYBEL29:def.5">Def5</a></i>
<br/>.= 
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c10">f2</font>
<b>by </b><i><a class="ref" href="waybel29.html#D5" target="_self" title="WAYBEL29:def.5">Def5</a></i>
;<br/><a NAME="E7:42_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel29.html#E3:42_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="waybel29.html#E2:42_1_1_2"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="funct_6.html#T87" title="FUNCT_6:th.87">FUNCT_6:87</a></i>;<br/><b>hence </b><a NAME="E8:42_1_1_2"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <b>by </b><i><a class="txt" href="waybel29.html#E3:42_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="waybel29.html#E2:42_1_1_2"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="funct_6.html#T87" title="FUNCT_6:th.87">FUNCT_6:87</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="E5:42_1_1"/>
 <a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a></span><br/>

<a NAME="E6:42_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</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"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">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" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c5">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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c5">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"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font></span>)</span> or <font color="Maroon" title="c5">x</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> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:42_1_1_3"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font></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="c5">x</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> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/>
<a NAME="E4:42_1_1_3"/><i><font color="Green" title="E22">A4</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel26.html#T33" title="WAYBEL26:th.33">WAYBEL26:33</a>, <a class="ref" href="yellow_1.html#T28" title="YELLOW_1:th.28">YELLOW_1:28</a></i>;<br/>
<a NAME="E5:42_1_1_3"/><b>then </b><i><font color="Green" title="E23">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T63" title="FUNCT_5:th.63">FUNCT_5:63</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <b>by </b><i><a class="txt" href="waybel29.html#E4:42_1_1_3"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g</font> =  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c7">x</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T34" title="WAYBEL26:th.34">WAYBEL26:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T2" title="WAYBEL26:th.2">WAYBEL26:2</a></i>;<br/>
<a NAME="E9:42_1_1_3"/>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E2:42_1_1_3"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="waybel29.html#E5:42_1_1_3"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="funct_6.html#T87" title="FUNCT_6:th.87">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" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</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="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="waybel_5.html#K7" title="WAYBEL_5:func.7">=&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel29.html#D5" target="_self" title="WAYBEL29:def.5">Def5</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:42_1_1_3"/>
<font color="Maroon" title="c5">x</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> <span class="p1">(<span class="default"><a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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="E7:42_1_1"/>
 <a href="waybel29.html#K5" title="WAYBEL29:func.5">commute</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">Y</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</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>
