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

<b>let </b><font color="Maroon">R1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">R2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:89"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R1</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R2</font> #)
 ;<br/>

<b>let </b><font color="Maroon">N1</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R1</font>;<br/><b>let </b><font color="Maroon">N2</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R2</font>;<br/>



<b>assume </b><a NAME="E2:89"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">N1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font>
 ;<br/>

<b>let </b><font color="Maroon">J1</font> be    <a href="yellow_6.html#M3">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N1</font>,<font color="Maroon">R1</font>;<br/><b>let </b><font color="Maroon">J2</font> be    <a href="yellow_6.html#M3">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N2</font>,<font color="Maroon">R2</font>;<br/>



<b>assume </b><a NAME="E3:89"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">J1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J2</font>
 ;<br/>

<a NAME="E4:89"/><i><font color="Green">E54</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J1</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>;<br/>
<a NAME="E5:89"/><i><font color="Green">E63</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J2</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>;<br/>
<a NAME="E6:89"/><b>then </b><i><font color="Green">E64</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</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="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:89"/><i><font color="Green">E88</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="waybel11.html#T2" target="_self">Th2</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">g</font> = the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span>;<br/>
<i><font color="Green">E89</font></i>:  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="waybel11.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E9:89"/>
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>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span> holds <br/>the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>proof </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="E1:89_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span>
 ;<br/>

<a NAME="E2:89_2"/><b>then </b>
<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"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span>
 
;<br/>
<a NAME="E3:89_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N1</font>, <font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J1</font></span>)</span><b> such that </b><br/><a NAME="E5:89_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N2</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J2</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>thus </b>the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="waybel11.html#T4" target="_self">Th4</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J1</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">x1</font></span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J2</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">y1</font></span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="waybel11.html#T4" target="_self">Th4</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:89"/><b>then </b><i><font color="Green">E91</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font></span>)</span> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E6:89"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>thus </b><a NAME="E11:89"/>
 <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J1</font> <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J2</font>
 <b>by </b><i>, <a class="txt" href="waybel11.html#E5:89"><i><font color="Green">E63</font></i></a>, <a class="ref" href="waybel11.html#T4" target="_self">Th4</a></i>;<br/>


</div>
