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

<b>let </b><font color="Maroon" title="c1">R1</font>, <font color="Maroon" title="c2">R2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:89"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">R2</font> #)
 ;<br/>

<b>let </b><font color="Maroon" title="c4">N1</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R1</font>;<br/><b>let </b><font color="Maroon" title="c5">N2</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">R2</font>;<br/>



<b>assume </b><a NAME="E2:89"/><i><font color="Green" title="E58">A1</font></i>: 
<font color="Maroon" title="c4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">J1</font> be    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</font>,<font color="Maroon" title="c1">R1</font>;<br/><b>let </b><font color="Maroon" title="c7">J2</font> be    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c2">R2</font>;<br/>



<b>assume </b><a NAME="E3:89"/><i><font color="Green" title="E59">A2</font></i>: 
<font color="Maroon" title="c6">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">J2</font>
 ;<br/>

<a NAME="E4:89"/><i><font color="Green" title="E60">A3</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c4">N1</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c6">J1</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></i>;<br/>
<a NAME="E5:89"/><i><font color="Green" title="E61">A4</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c5">N2</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c7">J2</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></i>;<br/>
<a NAME="E6:89"/><b>then </b><i><font color="Green" title="E62">A5</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="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</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="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:89"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="waybel11.html#E3:89"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="waybel11.html#E4:89"><i><font color="Green" title="E60">A3</font></i></a></i>;<br/>
<a NAME="E7:89"/><i><font color="Green" title="E63">A6</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:89"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="waybel11.html#E3:89"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="waybel11.html#E4:89"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="waybel11.html#E5:89"><i><font color="Green" title="E61">A4</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">f</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">g</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>;<br/>
<i><font color="Green" title="E64">A7</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E2:89"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="waybel11.html#E3:89"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="waybel11.html#E4:89"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="waybel11.html#E5:89"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E9:89"/>
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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> holds <br/>the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:89_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span>
 ;<br/>

<a NAME="E2:89_2"/><b>then </b>
<font color="Maroon" title="c10">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="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span>
 
;<br/>
<a NAME="E3:89_2"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c6">J1</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E4:89"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N1</font>, <font color="Maroon" title="c12">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c6">J1</font></span>)</span><b> such that </b><br/><a NAME="E5:89_2"/><i><font color="Green" title="E65">A8</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c11">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">N2</font> <b>by </b><i><a class="txt" href="waybel11.html#E2:89"><i><font color="Green" title="E58">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">y2</font> = <font color="Maroon" title="c12">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="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c7">J2</font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E2:89"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="waybel11.html#E3:89"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/>
<b>thus </b>the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font>
<b>by </b><i><a class="txt" href="waybel11.html#E5:89_2"><i><font color="Green" title="E65">A8</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">J1</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c11">x1</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c11">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">J2</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c13">y1</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y2</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c13">y1</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E3:89"><i><font color="Green" title="E59">A2</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">y1</font>,<font color="Maroon" title="c14">y2</font>
<b>by </b><i><a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="waybel11.html#E5:89_2"><i><font color="Green" title="E65">A8</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:89"/><b>then </b><i><font color="Green" title="E65">A9</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:89"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>thus </b><a NAME="E11:89"/>
 <a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c6">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c7">J2</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E6:89"><i><font color="Green" title="E62">A5</font></i></a>, <a class="txt" href="waybel11.html#E7:89"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="waybel11.html#E10:89"><i><font color="Green" title="E65">A9</font></i></a></i>;<br/>


</div>
