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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be  <a href="waybel_3.html#V4">non-Empty</a> <a href="waybel18.html#V1">TopSpace-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <font color="Maroon">J</font></span>)</span>;<br/>







<a NAME="E1:41"/><i><font color="Green">E45</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <font color="Maroon">J</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:41"/><i><font color="Green">E46</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel18.html#D3">WAYBEL18:def 3</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:41"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">I</font>,<span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</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="E1:41_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:41_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:41_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:41_1_1"/>
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:41_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">I</font>,<span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:41"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</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">I</font>,<span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:41"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">I</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>,<span class="p2">(<span class="default"><a href="card_3.html#K3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/>
<a NAME="E7:41"/><b>then </b><i><font color="Green">E66</font></i>: 
 ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">g</font> <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="card_3.html#K3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<a NAME="E8:41"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:41"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E11:41"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E12:41"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E13:41"/><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/><b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:41_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E5:41_2"/><i><font color="Green">E80</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel18.html#D4">WAYBEL18:def 4</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="waybel26.html#T1" target="_self">Th1</a>, , <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>
;<br/><b>hence </b><a NAME="E7:41_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="waybel26.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:41"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
