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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be    <a href="yellow_6.html#M4" title="YELLOW_6:mode.4">Convergence-Class</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>set </b><font color="Maroon" title="c3">CC</font> =  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font>;<br/>
<b>set </b><font color="Maroon" title="c4">CCC</font> =  <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>;<br/>
<a NAME="E1:121"/><i><font color="Green" title="E70">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="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</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 <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<a NAME="E2:121"/><i><font color="Green" title="E71">A2</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Olive" title="b1">N</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<a NAME="E3:121"/><i><font color="Green" title="E72">A3</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">n</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> holds <br/>for <font color="Olive" title="b3">X</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">N</font> holds  <font color="Olive" title="b3">X</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c6">n</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font>;<br/>



<b>assume </b><a NAME="E1:121_1"/><i><font color="Green" title="E73">A4</font></i>: 
<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">X</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">N</font>;<br/>

<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c5">N</font><b> such that </b><br/><a NAME="E3:121_1"/><i><font color="Green" title="E74">A5</font></i>: 
( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">X</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c5">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">N</font>  ex <font color="Olive" title="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">X</font> st <br/>for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">X</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b3">p</font> ) )
 <b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self" title="YELLOW_6:def.12">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">X</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c6">n</font> <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_1"><i><font color="Green" title="E73">A4</font></i></a></i>;<br/>
<a NAME="E6:121_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c9">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c6">n</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121_1"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_1"><i><font color="Green" title="E74">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:121_1"/>
<font color="Maroon" title="c7">X</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_1"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_1"><i><font color="Green" title="E74">A5</font></i></a>, <a class="ref" href="yellow_6.html#D12" target="_self" title="YELLOW_6:def.12">Def12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:121"/><i><font color="Green" title="E73">A6</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">n</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b2">n</font> holds  <font color="Olive" title="b3">x</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c6">n</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font>;<br/>



<b>assume </b><a NAME="E1:121_2"/><i><font color="Green" title="E74">A7</font></i>: 
<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">X</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c6">n</font>;<br/>

<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c6">n</font><b> such that </b><br/><a NAME="E3:121_2"/><i><font color="Green" title="E75">A8</font></i>: 
( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">X</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c6">n</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">n</font>  ex <font color="Olive" title="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">X</font> st <br/>for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">X</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b3">p</font> ) )
 <b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self" title="YELLOW_6:def.12">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">X</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">N</font> <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_2"><i><font color="Green" title="E74">A7</font></i></a></i>;<br/>
<a NAME="E6:121_2"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c9">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c5">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121_2"><i><font color="Green" title="E74">A7</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_2"><i><font color="Green" title="E75">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:121_2"/>
<font color="Maroon" title="c7">X</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_2"><i><font color="Green" title="E74">A7</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_2"><i><font color="Green" title="E75">A8</font></i></a>, <a class="ref" href="yellow_6.html#D12" target="_self" title="YELLOW_6:def.12">Def12</a></i>;<br/>


</div><b>end;</b></div>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:121_3"/><i><font color="Green" title="E74">A9</font></i>: 
 <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font>
 ;<br/><b>thus </b><a NAME="E2:121_3"/>
<font color="Maroon" title="c2">C</font> is <a href="yellow_6.html#V8" title="YELLOW_6:attr.8">topological</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:121_3_1"/>
<font color="Maroon" title="c2">C</font> is <a href="yellow_6.html#V4" title="YELLOW_6:attr.4">(CONSTANTS)</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D28" target="_self" title="YELLOW_6:def.28">YELLOW_6:def 28</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">N</font> be  <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D23" target="_self" title="YELLOW_6:def.23">YELLOW_6:def 23</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c5">N</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<a NAME="E2:121_3_1_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c5">N</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
;<br/>
<a NAME="E3:121_3_1_1"/><b>then </b>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c6">M</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">M</font> = <font color="Maroon" title="c6">M</font> as  <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="yellow_6.html#D6" target="_self" title="YELLOW_6:def.6">Def6</a></i>;<br/>
<i><font color="Green" title="E75">A10</font></i>:  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c7">M</font> = 
 <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_value_of</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">M</font>
<b>by </b><i><a class="ref" href="yellow_6.html#D10" target="_self" title="YELLOW_6:def.10">Def10</a></i>
<br/>.= 
 <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_value_of</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c5">N</font>

<br/>.= 
 <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c5">N</font>
<b>by </b><i><a class="ref" href="yellow_6.html#D10" target="_self" title="YELLOW_6:def.10">Def10</a></i>
;<br/>
<b>assume </b><a NAME="E6:121_3_1_1"/>
<font color="Maroon" title="c5">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">T</font>
 ;<br/>

<a NAME="E7:121_3_1_1"/><b>then </b>
<font color="Maroon" title="c7">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green" title="E38">Lm7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:121_3_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">N</font>,<span class="p2">(<span class="default"><a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c5">N</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green" title="E74">A9</font></i></a>, <a class="txt" href="yellow_6.html#E5:121_3_1_1"><i><font color="Green" title="E75">A10</font></i></a>, <a class="ref" href="yellow_6.html#D23" target="_self" title="YELLOW_6:def.23">Def23</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:121_3_1"/>
<font color="Maroon" title="c2">C</font> is <a href="yellow_6.html#V5" title="YELLOW_6:attr.5">(SUBNETS)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D24" target="_self" title="YELLOW_6:def.24">YELLOW_6:def 24</a><br/><b>let </b><font color="Maroon" title="c6">Y</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">N</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">M</font> = <font color="Maroon" title="c5">N</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c6">Y</font> as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c7">M</font> <b>by </b><i><a class="txt" href="yellow_6.html#E3:121"><i><font color="Green" title="E72">A3</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:121_3_1_2"/>
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">T</font>
 ;<br/>

<a NAME="E4:121_3_1_2"/><b>then </b><i><font color="Green" title="E75">A11</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green" title="E38">Lm7</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c9">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">p</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_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>assume </b><a NAME="E6:121_3_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ;<br/>

<a NAME="E7:121_3_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">M</font>,<font color="Maroon" title="c10">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green" title="E74">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:121_3_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green" title="E74">A9</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_3_1_2"><i><font color="Green" title="E75">A11</font></i></a>, <a class="ref" href="yellow_6.html#D24" target="_self" title="YELLOW_6:def.24">Def24</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:121_3_1"/>
<font color="Maroon" title="c2">C</font> is <a href="yellow_6.html#V6" title="YELLOW_6:attr.6">(DIVERGENCE)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">X</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D25" target="_self" title="YELLOW_6:def.25">YELLOW_6:def 25</a><br/><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">X</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c6">p</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_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>assume </b><a NAME="E3:121_3_1_3"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">T</font>
 ;<br/>

<a NAME="E4:121_3_1_3"/><b>then </b><i><font color="Green" title="E75">A12</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green" title="E38">Lm7</font></i></a></i>;<br/>
<b>assume </b><a NAME="E5:121_3_1_3"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c6">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c7">x</font><b> such that </b><br/><a NAME="E7:121_3_1_3"/><i><font color="Green" title="E76">A13</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <b>and </b><br/><a NAME="E8:121_3_1_3"/><i><font color="Green" title="E77">A14</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c9">y</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">z</font>,<font color="Maroon" title="c8">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green" title="E74">A9</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_3_1_3"><i><font color="Green" title="E75">A12</font></i></a>, <a class="ref" href="yellow_6.html#D25" target="_self" title="YELLOW_6:def.25">Def25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Y</font> = <font color="Maroon" title="c9">y</font> as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">X</font> <b>by </b><i><a class="txt" href="yellow_6.html#E4:121"><i><font color="Green" title="E73">A6</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">Y</font>
;<br/>

<b>thus </b><a NAME="E10:121_3_1_3"/>
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="yellow_6.html#E7:121_3_1_3"><i><font color="Green" title="E76">A13</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green" title="E38">Lm7</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">Z</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c10">Y</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c12">z</font> = <font color="Maroon" title="c11">Z</font> as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c9">y</font> <b>by </b><i><a class="txt" href="yellow_6.html#E3:121"><i><font color="Green" title="E72">A3</font></i></a></i>;<br/>
<a NAME="E12:121_3_1_3"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z</font>,<font color="Maroon" title="c8">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:121_3_1_3"><i><font color="Green" title="E77">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:121_3_1_3"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">Z</font>,<font color="Maroon" title="c6">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green" title="E74">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:121_3_1"/>
<font color="Maroon" title="c2">C</font> is <a href="yellow_6.html#V7" title="YELLOW_6:attr.7">(ITERATED_LIMITS)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">X</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D26" target="_self" title="YELLOW_6:def.26">YELLOW_6:def 26</a><br/><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">X</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c6">p</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_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>assume </b><a NAME="E3:121_3_1_4"/><i><font color="Green" title="E75">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c6">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">J</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">X</font>,<font color="Maroon" title="c1">T</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c10">I</font> = <font color="Maroon" title="c9">J</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x</font> ;<br/>
<a NAME="E5:121_3_1_4"/>
<font color="Maroon" title="c10">I</font> is    <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="c7">x</font>, <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</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="yellow_6.html#D15" target="_self" title="YELLOW_6:def.15">YELLOW_6:def 15</a><br/>

<b>assume </b><a NAME="E1:121_3_1_4_1"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">I</font>
 ;<br/>

<a NAME="E2:121_3_1_4_1"/><b>then </b>
<font color="Maroon" title="c11">i</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D15" target="_self" title="YELLOW_6:def.15">Def15</a></i>;<br/>
<b>hence </b><a NAME="E3:121_3_1_4_1"/>
<font color="Maroon" title="c11">i</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:121"><i><font color="Green" title="E71">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">I</font> = <font color="Maroon" title="c9">J</font> as    <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="c7">x</font>, <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> ;<br/>
<b>assume </b><a NAME="E7:121_3_1_4"/><i><font color="Green" title="E76">A16</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">X</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">x</font>;<br/><b>reconsider </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c12">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">X</font> ;<br/><a NAME="E2:121_3_1_4_2"/>
<font color="Maroon" title="c5">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c12">i</font>
 ;<br/><b>hence </b><a NAME="E3:121_3_1_4_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c12">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green" title="E74">A9</font></i></a>, <a class="txt" href="yellow_6.html#E7:121_3_1_4"><i><font color="Green" title="E76">A16</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:121_3_1_4"/><b>then </b><i><font color="Green" title="E77">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c11">I</font></span>)</span>,<font color="Maroon" title="c8">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green" title="E74">A9</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_3_1_4"><i><font color="Green" title="E75">A15</font></i></a>, <a class="ref" href="yellow_6.html#D26" target="_self" title="YELLOW_6:def.26">Def26</a></i>;<br/>
<i><font color="Green" title="E78">A18</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="c11">I</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="c11">I</font></span>)</span> #) = 
<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c5">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c9">J</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" target="_self" title="YELLOW_6:def.16">Def16</a></i>
<br/>.= 
 <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="c9">J</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="c9">J</font></span>)</span> #)
<b>by </b><i><a class="ref" href="yellow_6.html#D16" target="_self" title="YELLOW_6:def.16">Def16</a></i>
;<br/>
<a NAME="E11:121_3_1_4"/>
 <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="c11">I</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="c11">I</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="E12:121_3_1_4"/><b>then </b><i><font color="Green" title="E79">A19</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="c11">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c9">J</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E10:121_3_1_4"><i><font color="Green" title="E78">A18</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:121_3_1_4_4"/><i><font color="Green" title="E80">A20</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="c11">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">x</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c11">I</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self" title="YELLOW_6:th.35">Th35</a></i>;<br/><a NAME="E2:121_3_1_4_4"/><i><font color="Green" title="E81">A21</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="c9">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">X</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c9">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self" title="YELLOW_6:th.35">Th35</a></i>;<br/><b>assume </b><a NAME="E3:121_3_1_4_4"/>
<font color="Maroon" title="c12">j</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="c11">I</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c13">jj</font> = <font color="Maroon" title="c12">j</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_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c11">I</font></span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c14">j1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c15">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c11">I</font></span>)</span><b> such that </b><br/><a NAME="E6:121_3_1_4_4"/><i><font color="Green" title="E82">A22</font></i>: 
<font color="Maroon" title="c13">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">j1</font>,<font color="Maroon" title="c15">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3_1_4_4"><i><font color="Green" title="E80">A20</font></i></a>, <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="c16">j2</font> = <font color="Maroon" title="c15">j2</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="c11">I</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">i1</font> = <font color="Maroon" title="c14">j1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">X</font> ;<br/><b>set </b><font color="Maroon" title="c18">i2</font> = <font color="Maroon" title="c16">j2</font>;<br/><b>reconsider </b><font color="Maroon" title="c19">i</font> = <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c17">i1</font>,<font color="Maroon" title="c16">j2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c9">J</font></span>)</span> <b>by </b><i><a class="txt" href="yellow_6.html#E2:121_3_1_4_4"><i><font color="Green" title="E81">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</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="c11">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c11">I</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">jj</font>

<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="c11">I</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c14">j1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c14">j1</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121_3_1_4_4"><i><font color="Green" title="E82">A22</font></i></a>, <a class="ref" href="yellow_6.html#T36" target="_self" title="YELLOW_6:th.36">Th36</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="c9">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c17">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c9">J</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">i</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T36" target="_self" title="YELLOW_6:th.36">Th36</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="c9">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121_3_1_4_4"><i><font color="Green" title="E82">A22</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E14:121_3_1_4"/><b>then </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="c11">I</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="c9">J</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E12:121_3_1_4"><i><font color="Green" title="E79">A19</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E15:121_3_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c9">J</font></span>)</span>,<font color="Maroon" title="c6">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green" title="E74">A9</font></i></a>, <a class="txt" href="yellow_6.html#E9:121_3_1_4"><i><font color="Green" title="E77">A17</font></i></a>, <a class="txt" href="yellow_6.html#E10:121_3_1_4"><i><font color="Green" title="E78">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:121"/><i><font color="Green" title="E74">A23</font></i>: 
( <font color="Maroon" title="c2">C</font> is <a href="yellow_6.html#V4" title="YELLOW_6:attr.4">(CONSTANTS)</a> &amp; <font color="Maroon" title="c2">C</font> is <a href="yellow_6.html#V5" title="YELLOW_6:attr.5">(SUBNETS)</a> &amp; <font color="Maroon" title="c2">C</font> is <a href="yellow_6.html#V6" title="YELLOW_6:attr.6">(DIVERGENCE)</a> &amp; <font color="Maroon" title="c2">C</font> is <a href="yellow_6.html#V7" title="YELLOW_6:attr.7">(ITERATED_LIMITS)</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D28" target="_self" title="YELLOW_6:def.28">YELLOW_6:def 28</a><br/>

<b>then reconsider </b><font color="Maroon" title="c5">C'</font> = <font color="Maroon" title="c2">C</font> as  <a href="yellow_6.html#V8" title="YELLOW_6:attr.8">topological</a>  <a href="yellow_6.html#M4" title="YELLOW_6:mode.4">Convergence-Class</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#D28" target="_self" title="YELLOW_6:def.28">Def28</a></i>;<br/>
<a NAME="E8:121"/><i><font color="Green" title="E75">A24</font></i>: 
 <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a><br/><b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<a NAME="E1:121_4"/><i><font color="Green" title="E76">A25</font></i>: 
 <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <span class="p3">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21" target="_self" title="YELLOW_6:def.21">Def21</a></i>;<br/>
<b>assume </b><a NAME="E2:121_4"/><i><font color="Green" title="E77">A26</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">M</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>, <font color="Maroon" title="c9">p</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_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E4:121_4"/><i><font color="Green" title="E78">A27</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">M</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_4"><i><font color="Green" title="E76">A25</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E5:121_4"/><i><font color="Green" title="E79">A28</font></i>: 
<font color="Maroon" title="c8">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
;<br/>
<a NAME="E6:121_4"/>
 ex <font color="Olive" title="b1">N</font> being <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> st <br/>( <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">M</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</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#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_6.html#D14" target="_self" title="YELLOW_6:def.14">Def14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">M</font> = <font color="Maroon" title="c8">M</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">N</font> = <font color="Maroon" title="c10">M</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c9">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<a NAME="E10:121_4"/><i><font color="Green" title="E80">A29</font></i>: 
<font color="Maroon" title="c11">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="yellow_6.html#E5:121_4"><i><font color="Green" title="E79">A28</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green" title="E38">Lm7</font></i></a></i>;<br/>
<b>assume </b><a NAME="E11:121_4"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">Y</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c11">N</font><b> such that </b><br/><a NAME="E13:121_4"/><i><font color="Green" title="E81">A30</font></i>: 
<font color="Maroon" title="c13">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E14:121_4"/><i><font color="Green" title="E82">A31</font></i>: 
for <font color="Olive" title="b1">Z</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c13">Y</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">Z</font>,<font color="Maroon" title="c12">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E6:121"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="yellow_6.html#E10:121_4"><i><font color="Green" title="E80">A29</font></i></a>, <a class="ref" href="yellow_6.html#D25" target="_self" title="YELLOW_6:def.25">Def25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">YY</font> =  <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="c13">Y</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c13">Y</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="yellow_6.html#E15"><i><font color="Green" title="E7">Lm1</font></i></a>, <a class="txt" href="yellow_6.html#E16"><i><font color="Green" title="E8">Lm2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">X</font> = <font color="Maroon" title="c14">YY</font> <a href="yellow_6.html#K4" title="YELLOW_6:func.4">--&gt;</a> <font color="Maroon" title="c12">q</font>;<br/>
<a NAME="E16:121_4"/><i><font color="Green" title="E83">A32</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"><font color="Maroon" title="c14">YY</font> <a href="yellow_6.html#K4" title="YELLOW_6:func.4">--&gt;</a> <font color="Maroon" title="c12">q</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">YY</font> <a href="yellow_6.html#K4" title="YELLOW_6:func.4">--&gt;</a> <font color="Maroon" title="c12">q</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">YY</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D7" target="_self" title="YELLOW_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">X</font> = <font color="Maroon" title="c14">YY</font> <a href="yellow_6.html#K4" title="YELLOW_6:func.4">--&gt;</a> <font color="Maroon" title="c12">q</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E18:121_4"/><i><font color="Green" title="E84">A33</font></i>: 
 ex <font color="Olive" title="b1">N0</font> being <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">N0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Y</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E13:121_4"><i><font color="Green" title="E81">A30</font></i></a>, <a class="ref" href="yellow_6.html#D14" target="_self" title="YELLOW_6:def.14">Def14</a></i>;<br/>
<a NAME="E19:121_4"/>
 <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="c16">X</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c16">X</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="c13">Y</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c13">Y</font> #)
 
<b>by </b><i><a class="ref" href="yellow_6.html#D7" target="_self" title="YELLOW_6:def.7">Def7</a></i>;<br/>
<a NAME="E20:121_4"/><b>then </b><i><font color="Green" title="E85">A34</font></i>: 
<font color="Maroon" title="c16">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E18:121_4"><i><font color="Green" title="E84">A33</font></i></a>, <a class="ref" href="yellow_6.html#D14" target="_self" title="YELLOW_6:def.14">Def14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">X'</font> = <font color="Maroon" title="c16">X</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<a NAME="E22:121_4"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c16">X</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
;<br/>
<a NAME="E23:121_4"/><b>then </b>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c17">X'</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">X'</font> = <font color="Maroon" title="c17">X'</font> as  <a href="yellow_6.html#V1" title="YELLOW_6:attr.1">constant</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="ref" href="yellow_6.html#D6" target="_self" title="YELLOW_6:def.6">Def6</a></i>;<br/>
<a NAME="E25:121_4"/>
 <a href="yellow_6.html#K5" title="YELLOW_6:func.5">the_value_of</a> <font color="Maroon" title="c16">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T22" target="_self" title="YELLOW_6:th.22">Th22</a></i>;<br/>
<a NAME="E26:121_4"/><b>then </b><i><font color="Green" title="E86">A35</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">X</font>,<font color="Maroon" title="c12">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="yellow_6.html#E20:121_4"><i><font color="Green" title="E85">A34</font></i></a>, <a class="ref" href="yellow_6.html#D23" target="_self" title="YELLOW_6:def.23">Def23</a></i>;<br/>
<a NAME="E27:121_4"/><i><font color="Green" title="E87">A36</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c10">M</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4"><i><font color="Green" title="E77">A26</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4"><i><font color="Green" title="E78">A27</font></i></a>, <a class="ref" href="yellow_6.html#D22" target="_self" title="YELLOW_6:def.22">Def22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">Y'</font> = <font color="Maroon" title="c13">Y</font> as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c10">M</font> <b>by </b><i><a class="txt" href="yellow_6.html#E3:121"><i><font color="Green" title="E72">A3</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> ex <font color="Olive" title="b2">Ji</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">Ji</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b2">Ji</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> where <font color="Olive" title="b3">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> </span>}</span>  );<br/>
<a NAME="E29:121_4"/><i><font color="Green" title="E88">A37</font></i>: 
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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">X</font> holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:121_4_1"/>
<font color="Maroon" title="c20">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 <font color="Maroon" title="c16">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c21">i'</font> = <font color="Maroon" title="c20">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c19">Y'</font> <b>by </b><i><a class="ref" href="yellow_6.html#T13" target="_self" title="YELLOW_6:th.13">Th13</a></i>;<br/>
<a NAME="E3:121_4_1"/>
 <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c10">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c19">Y'</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T41" target="_self" title="YELLOW_6:th.41">Th41</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span><b> such that </b><br/><a NAME="E5:121_4_1"/><i><font color="Green" title="E89">A38</font></i>: 
<font color="Maroon" title="c22">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c19">Y'</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">c</font></span>)</span> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c19">Y'</font> : <font color="Maroon" title="c21">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> </span>}</span> 
 <b>and </b><br/><a NAME="E6:121_4_1"/><i><font color="Green" title="E90">A39</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c22">S</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E27:121_4"><i><font color="Green" title="E87">A36</font></i></a>, <a class="ref" href="yellow_6.html#T43" target="_self" title="YELLOW_6:th.43">Th43</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">Go</font> being   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c5">C'</font><b> such that </b><br/><a NAME="E8:121_4_1"/><i><font color="Green" title="E91">A40</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">Go</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C'</font>
 <b>and </b><br/><a NAME="E9:121_4_1"/><i><font color="Green" title="E92">A41</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c23">Go</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">S</font>
 <b>and </b><br/><a NAME="E10:121_4_1"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c23">Go</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E6:121_4_1"><i><font color="Green" title="E90">A39</font></i></a>, <a class="ref" href="yellow_6.html#T52" target="_self" title="YELLOW_6:th.52">Th52</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">Ji</font> = <font color="Maroon" title="c23">Go</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">i</font> = <font color="Maroon" title="c21">i'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c24">Ji</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c25">i</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c24">Ji</font>
;<br/>

<b>thus </b><a NAME="E13:121_4_1"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">i</font> &amp; <font color="Maroon" title="c24">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">Ji</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">Ji</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )
 <b>by </b><i><a class="txt" href="yellow_6.html#E8:121_4_1"><i><font color="Green" title="E91">A40</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c26">e</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><br/>

<b>assume </b><a NAME="E14:121_4_1"/>
<font color="Maroon" title="c26">e</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> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c24">Ji</font>
 ;<br/>

<a NAME="E15:121_4_1"/><b>then </b>
<font color="Maroon" title="c26">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">S</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E9:121_4_1"><i><font color="Green" title="E92">A41</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c27">c'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c19">Y'</font><b> such that </b><br/><a NAME="E17:121_4_1"/><i><font color="Green" title="E93">A42</font></i>: 
<font color="Maroon" title="c26">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Y'</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c27">c'</font>
 <b>and </b><br/><a NAME="E18:121_4_1"/><i><font color="Green" title="E94">A43</font></i>: 
<font color="Maroon" title="c21">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">c'</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E5:121_4_1"><i><font color="Green" title="E89">A38</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">cc</font> = <font color="Maroon" title="c27">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> ;<br/>
<a NAME="E20:121_4_1"/>
<font color="Maroon" title="c26">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c28">cc</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E17:121_4_1"><i><font color="Green" title="E93">A42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:121_4_1"/>
<font color="Maroon" title="c26">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">c</font></span>)</span> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Maroon" title="c25">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E18:121_4_1"><i><font color="Green" title="E94">A43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c20">J</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">X</font><b> such that </b><br/><a NAME="E31:121_4"/><i><font color="Green" title="E89">A44</font></i>: 
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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c20">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="yellow_6.html#E29:121_4"><i><font color="Green" title="E88">A37</font></i></a>);<br/></i>
<div><i><font color="Green" title="E90">A45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:121_4_2"/>
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">J</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c22">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:121_4_2"/><i><font color="Green" title="E91">A46</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c20">J</font>
 <b>and </b><br/><a NAME="E4:121_4_2"/><i><font color="Green" title="E92">A47</font></i>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:121_4_2"/>
<font color="Maroon" title="c22">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 <font color="Maroon" title="c16">X</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:121_4_2"><i><font color="Green" title="E91">A46</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:121_4_2"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> ex <font color="Olive" title="b2">Ji</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">Ji</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b2">Ji</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> where <font color="Olive" title="b3">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green" title="E89">A44</font></i></a></i>;<br/><b>hence </b><a NAME="E7:121_4_2"/>
<font color="Maroon" title="c21">y</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E4:121_4_2"><i><font color="Green" title="E92">A47</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E91">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:121_4_3"/>
<font color="Maroon" title="c21">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 <font color="Maroon" title="c16">X</font>
 ;<br/><a NAME="E2:121_4_3"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> ex <font color="Olive" title="b2">Ji</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">Ji</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b2">Ji</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> where <font color="Olive" title="b3">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green" title="E89">A44</font></i></a></i>;<br/><b>hence </b><a NAME="E3:121_4_3"/>
<font color="Maroon" title="c20">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:121_4_4"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c20">J</font>
 ;<br/><a NAME="E2:121_4_4"/><b>then </b>
<font color="Maroon" title="c21">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 <font color="Maroon" title="c16">X</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:121_4_4"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> ex <font color="Olive" title="b2">Ji</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">Ji</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b2">Ji</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> where <font color="Olive" title="b3">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green" title="E89">A44</font></i></a></i>;<br/><b>hence </b><a NAME="E4:121_4_4"/>
<font color="Maroon" title="c20">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> is    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E35:121_4"/><b>then </b>
<font color="Maroon" title="c20">J</font> is <a href="pralg_1.html#V2" title="PRALG_1:attr.2">1-sorted-yielding</a>
 
<b>by </b><i><a class="ref" href="pralg_1.html#D11" title="PRALG_1:def.11">PRALG_1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">J</font> = <font color="Maroon" title="c20">J</font> as  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">yielding_non-empty_carriers</a>  <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="c16">X</font>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="yellow_6.html#E32:121_4"><i><font color="Green" title="E90">A45</font></i></a>, <a class="txt" href="yellow_6.html#E33:121_4"><i><font color="Green" title="E91">A48</font></i></a>, <a class="ref" href="yellow_6.html#D4" target="_self" title="YELLOW_6:def.4">Def4</a>, <a class="ref" href="yellow_6.html#T33" target="_self" title="YELLOW_6:th.33">Th33</a></i>;<br/>
<a NAME="E37:121_4"/>
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font>;<br/>

<a NAME="E1:121_4_5"/>
 ex <font color="Olive" title="b1">ii</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> ex <font color="Olive" title="b2">Ji</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ii</font> &amp; <font color="Olive" title="b2">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c22">i</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">Ji</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b2">Ji</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> where <font color="Olive" title="b3">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Olive" title="b1">ii</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green" title="E89">A44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:121_4_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c22">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c22">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T14" target="_self" title="YELLOW_6:th.14">Th14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:121_4"/><b>then </b><i><font color="Green" title="E92">A49</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c21">J</font></span>)</span>,<font color="Maroon" title="c12">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121"><i><font color="Green" title="E74">A23</font></i></a>, <a class="txt" href="yellow_6.html#E26:121_4"><i><font color="Green" title="E86">A35</font></i></a>, <a class="ref" href="yellow_6.html#D26" target="_self" title="YELLOW_6:def.26">Def26</a></i>;<br/>
<a NAME="E39:121_4"/><i><font color="Green" title="E93">A50</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="c21">J</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="c21">J</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="c16">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c21">J</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" target="_self" title="YELLOW_6:def.16">Def16</a></i>;<br/>
<a NAME="E40:121_4"/><b>then </b><i><font color="Green" title="E94">A51</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="c21">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c16">X</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="c21">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E41:121_4"/>
 <a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c21">J</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c13">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : $1 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c22">B</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">Y</font><b> such that </b><br/><a NAME="E2:121_4_6"/><i><font color="Green" title="E95">A52</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> holds  <font color="Maroon" title="c22">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:121_4_6_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">B</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c23">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:121_4_6_1"/><i><font color="Green" title="E96">A53</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">B</font>
 <b>and </b><br/><a NAME="E4:121_4_6_1"/><i><font color="Green" title="E97">A54</font></i>: 
<font color="Maroon" title="c22">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">i</font> = <font color="Maroon" title="c23">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> <b>by </b><i><a class="txt" href="yellow_6.html#E3:121_4_6_1"><i><font color="Green" title="E96">A53</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c25">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font><b> such that </b><br/><a NAME="E7:121_4_6_1"/><i><font color="Green" title="E98">A55</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <b>and </b><br/><a NAME="E8:121_4_6_1"/>
<font color="Maroon" title="c24">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D5" target="_self" title="YELLOW_6:def.5">Def5</a></i>;<br/><a NAME="E9:121_4_6_1"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Maroon" title="c24">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6_1"><i><font color="Green" title="E98">A55</font></i></a></i>;<br/><b>hence </b><a NAME="E10:121_4_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6"><i><font color="Green" title="E95">A52</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4_6_1"><i><font color="Green" title="E97">A54</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c23">B</font> = <font color="Maroon" title="c22">B</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">Y</font> <b>by </b><i><a class="ref" href="relat_1.html#D9" title="RELAT_1:def.9">RELAT_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">B'</font> = <font color="Maroon" title="c23">B</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">X</font> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green" title="E83">A32</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = 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="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c25">M</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">X</font><b> such that </b><br/><a NAME="E7:121_4_6"/><i><font color="Green" title="E96">A56</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font> holds  <font color="Maroon" title="c25">M</font> <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> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ex <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; 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="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 <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 <font color="Maroon" title="c13">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E8:121_4_6"/><i><font color="Green" title="E97">A57</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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 <font color="Maroon" title="c16">X</font> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/> ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">i</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:121_4_6_2"/><i><font color="Green" title="E98">A58</font></i>: 
<font color="Maroon" title="c26">i</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 <font color="Maroon" title="c16">X</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c27">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:121_4_6_2"/><i><font color="Green" title="E99">A59</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font>, <font color="Maroon" title="c29">Ji</font> being   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:121_4_6_2"/><i><font color="Green" title="E100">A60</font></i>: 
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">e</font>
 <b>and </b><br/><a NAME="E5:121_4_6_2"/><i><font color="Green" title="E101">A61</font></i>: 
<font color="Maroon" title="c29">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 <b>and </b><br/><a NAME="E6:121_4_6_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">Ji</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>and </b><br/><a NAME="E7:121_4_6_2"/><i><font color="Green" title="E102">A62</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c29">Ji</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">c</font></span>)</span> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Maroon" title="c28">e</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green" title="E89">A44</font></i></a>, <a class="txt" href="yellow_6.html#E1:121_4_6_2"><i><font color="Green" title="E98">A58</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c30">i'</font> = <font color="Maroon" title="c26">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font> <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_4_6_2"><i><font color="Green" title="E98">A58</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c29">Ji</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <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 <font color="Maroon" title="c13">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2;<br/>
<a NAME="E9:121_4_6_2"/><i><font color="Green" title="E103">A63</font></i>: 
for <font color="Olive" title="b1">ji</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c29">Ji</font>  ex <font color="Olive" title="b2">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c24">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">i'</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">ji</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c31">ji</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c29">Ji</font>;<br/>

<a NAME="E1:121_4_6_2_1"/>
<font color="Maroon" title="c31">ji</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 <font color="Maroon" title="c29">Ji</font>
 
;<br/>
<a NAME="E2:121_4_6_2_1"/><b>then </b>
<font color="Maroon" title="c31">ji</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 <font color="Maroon" title="c29">Ji</font>
 
<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="E3:121_4_6_2_1"/><b>then </b>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c29">Ji</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">ji</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> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c29">Ji</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:121_4_6_2_1"/><b>then </b>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c29">Ji</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">ji</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">c</font></span>)</span> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Maroon" title="c28">e</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6_2"><i><font color="Green" title="E102">A62</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c32">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font><b> such that </b><br/><a NAME="E6:121_4_6_2_1"/><i><font color="Green" title="E104">A64</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c29">Ji</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c32">c</font>
 <b>and </b><br/><a NAME="E7:121_4_6_2_1"/><i><font color="Green" title="E105">A65</font></i>: 
<font color="Maroon" title="c28">e</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c32">c</font>
 ;<br/>
<a NAME="E8:121_4_6_2_1"/>
<font color="Maroon" title="c32">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">cc</font> where <font color="Olive" title="b1">cc</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Maroon" title="c28">e</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">cc</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6_2_1"><i><font color="Green" title="E105">A65</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c33">c</font> = <font color="Maroon" title="c32">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c24">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">i'</font> <b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6"><i><font color="Green" title="E95">A52</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4_6_2"><i><font color="Green" title="E100">A60</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c33">c</font>
;<br/>

<b>thus </b><a NAME="E10:121_4_6_2_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c29">Ji</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">ji</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 <font color="Maroon" title="c13">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">c</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E6:121_4_6_2_1"><i><font color="Green" title="E104">A64</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c31">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c29">Ji</font>,<font color="Maroon" title="c24">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">i'</font><b> such that </b><br/><a NAME="E11:121_4_6_2"/><i><font color="Green" title="E104">A66</font></i>: 
for <font color="Olive" title="b1">ji</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c29">Ji</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">ji</font>,<font color="Maroon" title="c31">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">ji</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="yellow_6.html#E9:121_4_6_2"><i><font color="Green" title="E103">A63</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c32">f</font> = <font color="Maroon" title="c31">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c29">Ji</font>,<font color="Maroon" title="c23">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c33">ji</font> = <font color="Maroon" title="c27">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c29">Ji</font> <b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green" title="E96">A56</font></i></a>, <a class="txt" href="yellow_6.html#E1:121_4_6_2"><i><font color="Green" title="E98">A58</font></i></a>, <a class="txt" href="yellow_6.html#E2:121_4_6_2"><i><font color="Green" title="E99">A59</font></i></a>, <a class="txt" href="yellow_6.html#E5:121_4_6_2"><i><font color="Green" title="E101">A61</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font>
;<br/>

<a NAME="E14:121_4_6_2"/>
<font color="Maroon" title="c32">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">ji</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 
;<br/>
<b>hence </b><a NAME="E15:121_4_6_2"/>
<font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 ;<br/>

<b>take </b>
<font color="Maroon" title="c32">f</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c30">i'</font>
;<br/>

<b>thus </b><a NAME="E16:121_4_6_2"/>
( <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> &amp; <font color="Maroon" title="c30">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">i</font> )
 ;<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"><font color="Maroon" title="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c30">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> = 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c29">Ji</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">ji</font>
<b>by </b><i><a class="txt" href="yellow_6.html#E5:121_4_6_2"><i><font color="Green" title="E101">A61</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c13">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E11:121_4_6_2"><i><font color="Green" title="E104">A66</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c26">u</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c25">M</font>,<font color="Maroon" title="c24">B'</font><b> such that </b><br/><a NAME="E10:121_4_6"/><i><font color="Green" title="E98">A67</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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 <font color="Maroon" title="c16">X</font> holds <br/> ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c24">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> st <br/>( <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">i</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1" title="MSSUBFAM:sch.1">MSSUBFAM:sch 1</a>(<a class="txt" href="yellow_6.html#E8:121_4_6"><i><font color="Green" title="E97">A57</font></i></a>);<br/></i>
<a NAME="E11:121_4_6"/><i><font color="Green" title="E99">A68</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font><br/> for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> holds  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="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</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 <font color="Maroon" title="c13">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font>;<br/><b>let </b><font color="Maroon" title="c28">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>;<br/>



<b>consider </b><font color="Maroon" title="c29">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>,<font color="Maroon" title="c24">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font><b> such that </b><br/><a NAME="E2:121_4_6_3"/><i><font color="Green" title="E100">A69</font></i>: 
<font color="Maroon" title="c29">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 <b>and </b><br/><a NAME="E3:121_4_6_3"/><i><font color="Green" title="E101">A70</font></i>: 
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> <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c29">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c27">i</font>]
 <b>by </b><i><a class="txt" href="yellow_6.html#E10:121_4_6"><i><font color="Green" title="E98">A67</font></i></a></i>;<br/>
<a NAME="E4:121_4_6_3"/>
<font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <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"><font color="Maroon" title="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c27">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green" title="E96">A56</font></i></a></i>;<br/>
<a NAME="E5:121_4_6_3"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c29">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font>,<font color="Maroon" title="c28">j</font>,<font color="Maroon" title="c27">i</font>]
 
<b>by </b><i><a class="txt" href="yellow_6.html#E3:121_4_6_3"><i><font color="Green" title="E101">A70</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:121_4_6_3"/>
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="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c27">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</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 <font color="Maroon" title="c13">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6_3"><i><font color="Green" title="E100">A69</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font>,  <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="c21">J</font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$2 <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> $1</span>)</span>;<br/>
<a NAME="E12:121_4_6"/><i><font color="Green" title="E100">A71</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font><br/> for <font color="Olive" title="b2">y</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="c21">J</font></span>)</span> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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 <font color="Maroon" title="c13">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font>;<br/><b>let </b><font color="Maroon" title="c28">y</font> be   <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="c21">J</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon" title="c29">k</font> = <font color="Maroon" title="c27">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green" title="E83">A32</font></i></a></i>;<br/>
<a NAME="E2:121_4_6_4"/><i><font color="Green" title="E101">A72</font></i>: 
<font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font>,<font color="Maroon" title="c23">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font>] <b>means </b><font color="Maroon" title="c29">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> $1;<br/>
<b>set </b><font color="Maroon" title="c30">ZZ</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] </span>}</span> ;<br/>
<a NAME="E3:121_4_6_4"/><i><font color="Green" title="E102">A73</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c13">Y</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E4:121_4_6_4"/><i><font color="Green" title="E103">A74</font></i>: 
<font color="Maroon" title="c23">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6"><i><font color="Green" title="E95">A52</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">x'</font> = <font color="Maroon" title="c27">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c18">X'</font> ;<br/>
<a NAME="E6:121_4_6_4"/>
<font color="Maroon" title="c28">y</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_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c21">J</font></span>)</span>
 
;<br/>
<a NAME="E7:121_4_6_4"/><b>then </b><i><font color="Green" title="E104">A75</font></i>: 
<font color="Maroon" title="c28">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E8:121_4_6_4"/>
<font color="Maroon" title="c31">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 <font color="Maroon" title="c18">X'</font>
 
;<br/>
<a NAME="E9:121_4_6_4"/><b>then </b>
<font color="Maroon" title="c31">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:121_4_6_4"/><b>then </b>
<font color="Maroon" title="c28">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x'</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6_4"><i><font color="Green" title="E104">A75</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E11:121_4_6_4"/><b>then </b>
<font color="Maroon" title="c28">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">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"><font color="Maroon" title="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c27">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self" title="YELLOW_6:th.9">Th9</a></i>;<br/>
<a NAME="E12:121_4_6_4"/><b>then </b>
<font color="Maroon" title="c28">y</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green" title="E96">A56</font></i></a></i>;<br/>
<a NAME="E13:121_4_6_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">y</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c27">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6_4"><i><font color="Green" title="E101">A72</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E14:121_4_6_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">y</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c27">x</font></span>)</span> <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 <font color="Maroon" title="c13">Y</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:121_4_6_4"><i><font color="Green" title="E102">A73</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4_6_4"><i><font color="Green" title="E103">A74</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c27">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c16">X</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="c21">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">Y</font><b> such that </b><br/><a NAME="E14:121_4_6"/><i><font color="Green" title="E101">A76</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font><br/> for <font color="Olive" title="b2">y</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="c21">J</font></span>)</span> holds  <font color="Maroon" title="c27">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="funct_7.html#S1" title="FUNCT_7:sch.1">FUNCT_7:sch 1</a>(<a class="txt" href="yellow_6.html#E12:121_4_6"><i><font color="Green" title="E100">A71</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c28">f</font> = <font color="Maroon" title="c27">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="c21">J</font></span>)</span>,<font color="Maroon" title="c13">Y</font> <b>by </b><i><a class="txt" href="yellow_6.html#E40:121_4"><i><font color="Green" title="E94">A51</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c28">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12" target="_self" title="YELLOW_6:def.12">YELLOW_6:def 12</a><br/>

<b>set </b><font color="Maroon" title="c29">h</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="c21">J</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c30">g</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c19">Y'</font>;<br/>
<a NAME="E16:121_4_6"/><i><font color="Green" title="E102">A77</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <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="c21">J</font></span>)</span> iff ( <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> <font color="Maroon" title="c28">f</font> &amp; <font color="Maroon" title="c28">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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 <font color="Maroon" title="c19">Y'</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c31">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:121_4_6_5_1"/>
<font color="Maroon" title="c31">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="c21">J</font></span>)</span>
 ;<br/><a NAME="E2:121_4_6_5_1"/><b>then </b>
<font color="Maroon" title="c31">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="c21">J</font></span>)</span>
 ;<br/><a NAME="E3:121_4_6_5_1"/><b>then </b>
<font color="Maroon" title="c31">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="c18">X'</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self" title="YELLOW_6:th.35">Th35</a></i>;<br/><a NAME="E4:121_4_6_5_1"/><b>then </b><i><font color="Green" title="E103">A78</font></i>: 
<font color="Maroon" title="c31">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="c18">X'</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="c21">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:121_4_6_5_1"/>
<font color="Maroon" title="c31">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> <font color="Maroon" title="c28">f</font>
 <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="E6:121_4_6_5_1"/>
<font color="Maroon" title="c28">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">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 <font color="Maroon" title="c13">Y</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E4:121_4_6_5_1"><i><font color="Green" title="E103">A78</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E7:121_4_6_5_1"/>
<font color="Maroon" title="c28">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">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 <font color="Maroon" title="c19">Y'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:121_4_6_5"/>
( <font color="Maroon" title="c31">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> <font color="Maroon" title="c28">f</font> &amp; <font color="Maroon" title="c28">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">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 <font color="Maroon" title="c19">Y'</font> )
 ;<br/>

<a NAME="E3:121_4_6_5"/><b>then </b>
<font color="Maroon" title="c31">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="c18">X'</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="c21">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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="E4:121_4_6_5"/><b>then </b>
<font color="Maroon" title="c31">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="c18">X'</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E5:121_4_6_5"/><b>then </b>
<font color="Maroon" title="c31">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="c21">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self" title="YELLOW_6:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E6:121_4_6_5"/>
<font color="Maroon" title="c31">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="c21">J</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/>


</div><b>end;</b></div>
<a NAME="E17:121_4_6"/>
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="c21">J</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="c21">J</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 <font color="Maroon" title="c19">Y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c31">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:121_4_6_6"/>
<font color="Maroon" title="c31">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="c21">J</font></span>)</span>
 ;<br/>

<a NAME="E2:121_4_6_6"/><b>then </b>
<font color="Maroon" title="c31">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="c21">J</font></span>)</span>
 
;<br/>
<a NAME="E3:121_4_6_6"/><b>then </b>
<font color="Maroon" title="c31">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="c18">X'</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self" title="YELLOW_6:th.35">Th35</a></i>;<br/>
<a NAME="E4:121_4_6_6"/><b>then </b>
<font color="Maroon" title="c31">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="c18">X'</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="c21">J</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c32">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c18">X'</font>, <font color="Maroon" title="c33">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="c21">J</font></span>)</span><b> such that </b><br/><a NAME="E6:121_4_6_6"/><i><font color="Green" title="E103">A79</font></i>: 
<font color="Maroon" title="c31">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="c32">x1</font>,<font color="Maroon" title="c33">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="c34">x'</font> = <font color="Maroon" title="c32">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font> ;<br/>
<a NAME="E8:121_4_6_6"/><i><font color="Green" title="E104">A80</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</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 <font color="Maroon" title="c18">X'</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:121_4_6_6"/>
<font color="Maroon" title="c33">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="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c21">J</font></span>)</span>
 
;<br/>
<a NAME="E10:121_4_6_6"/><b>then </b><i><font color="Green" title="E105">A81</font></i>: 
<font color="Maroon" title="c33">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E11:121_4_6_6"/>
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="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c34">x'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x1</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self" title="YELLOW_6:th.9">Th9</a></i>;<br/>
<a NAME="E12:121_4_6_6"/><b>then </b>
<font color="Maroon" title="c33">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">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"><font color="Maroon" title="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c34">x'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:121_4_6_6"><i><font color="Green" title="E104">A80</font></i></a>, <a class="txt" href="yellow_6.html#E10:121_4_6_6"><i><font color="Green" title="E105">A81</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c35">j</font> = <font color="Maroon" title="c33">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x'</font> <b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green" title="E96">A56</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="c21">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">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="c21">J</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c32">x1</font>,<font color="Maroon" title="c33">x2</font>
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121_4_6_6"><i><font color="Green" title="E103">A79</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="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c34">x'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_6.html#D16" target="_self" title="YELLOW_6:def.16">Def16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c19">Y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">j</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E11:121_4_6"><i><font color="Green" title="E99">A68</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c19">Y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c32">x1</font>,<font color="Maroon" title="c33">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E14:121_4_6"><i><font color="Green" title="E101">A76</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c19">Y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121_4_6_6"><i><font color="Green" title="E103">A79</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:121_4_6"/>
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="c21">J</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c13">Y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">f</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4_6"><i><font color="Green" title="E102">A77</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>

<b>let </b><font color="Maroon" title="c31">m</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font>;<br/>

<b>consider </b><font color="Maroon" title="c32">F</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="c21">J</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">n</font> = <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c31">m</font>,<font color="Maroon" title="c32">F</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c21">J</font></span>)</span> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green" title="E83">A32</font></i></a>, <a class="txt" href="yellow_6.html#E40:121_4"><i><font color="Green" title="E94">A51</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">F</font> = <font color="Maroon" title="c32">F</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="c21">J</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c35">m'</font> = <font color="Maroon" title="c31">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green" title="E83">A32</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c33">n</font>
;<br/>

<b>let </b><font color="Maroon" title="c36">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c21">J</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c37">k'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font>, <font color="Maroon" title="c38">G</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="c21">J</font></span>)</span><b> such that </b><br/><a NAME="E24:121_4_6"/><i><font color="Green" title="E103">A82</font></i>: 
<font color="Maroon" title="c36">p</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="c37">k'</font>,<font color="Maroon" title="c38">G</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E40:121_4"><i><font color="Green" title="E94">A51</font></i></a>, <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="c39">k</font> = <font color="Maroon" title="c37">k'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green" title="E83">A32</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c40">k''</font> = <font color="Maroon" title="c37">k'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c18">X'</font> ;<br/>
<a NAME="E27:121_4_6"/><i><font color="Green" title="E104">A83</font></i>: 
<font color="Maroon" title="c28">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c37">k'</font>,<font color="Maroon" title="c38">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E14:121_4_6"><i><font color="Green" title="E101">A76</font></i></a></i>;<br/>
<a NAME="E28:121_4_6"/><i><font color="Green" title="E105">A84</font></i>: 
<font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font>,<font color="Maroon" title="c23">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>then </b><i><font color="Green" title="E106">A85</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font></span>)</span> = 
<font color="Maroon" title="c25">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
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="c21">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c37">k'</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green" title="E96">A56</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">k''</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self" title="YELLOW_6:th.9">Th9</a></i>
;<br/>
<a NAME="E30:121_4_6"/><i><font color="Green" title="E107">A86</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</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 <font color="Maroon" title="c18">X'</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E31:121_4_6"/>
<font color="Maroon" title="c38">G</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_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c21">J</font></span>)</span>
 
;<br/>
<a NAME="E32:121_4_6"/><b>then </b>
<font color="Maroon" title="c38">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c21">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E33:121_4_6"/><b>then </b>
<font color="Maroon" title="c38">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">k''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E29:121_4_6"><i><font color="Green" title="E106">A85</font></i></a>, <a class="txt" href="yellow_6.html#E30:121_4_6"><i><font color="Green" title="E107">A86</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E34:121_4_6"/><b>then </b><i><font color="Green" title="E108">A87</font></i>: 
<font color="Maroon" title="c28">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c36">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E27:121_4_6"><i><font color="Green" title="E104">A83</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="txt" href="yellow_6.html#E24:121_4_6"><i><font color="Green" title="E103">A82</font></i></a></i>;<br/>
<a NAME="E35:121_4_6"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E28:121_4_6"><i><font color="Green" title="E105">A84</font></i></a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E36:121_4_6"/><b>then </b>
<font color="Maroon" title="c28">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c36">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">k</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E34:121_4_6"><i><font color="Green" title="E108">A87</font></i></a></i>;<br/>
<a NAME="E37:121_4_6"/><b>then </b>
<font color="Maroon" title="c28">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c36">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font> : <font color="Maroon" title="c39">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6"><i><font color="Green" title="E95">A52</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c41">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">Y</font><b> such that </b><br/><a NAME="E39:121_4_6"/><i><font color="Green" title="E109">A88</font></i>: 
<font color="Maroon" title="c41">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c36">p</font>
 <b>and </b><br/><a NAME="E40:121_4_6"/><i><font color="Green" title="E110">A89</font></i>: 
<font color="Maroon" title="c39">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c41">c</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c42">G</font> = <font color="Maroon" title="c38">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="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c21">J</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c43">k'</font> = <font color="Maroon" title="c39">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">X</font> ;<br/>
<b>assume </b><a NAME="E43:121_4_6"/>
<font color="Maroon" title="c33">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c36">p</font>
 ;<br/>

<a NAME="E44:121_4_6"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c35">m'</font>,<font color="Maroon" title="c34">F</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c43">k'</font>,<font color="Maroon" title="c42">G</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E39:121_4"><i><font color="Green" title="E93">A50</font></i></a>, <a class="txt" href="yellow_6.html#E24:121_4_6"><i><font color="Green" title="E103">A82</font></i></a>, <a class="txt" href="yellow_6.html#E20"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/>
<a NAME="E45:121_4_6"/><b>then </b>
<font color="Maroon" title="c35">m'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c43">k'</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>
<a NAME="E46:121_4_6"/><b>then </b>
<font color="Maroon" title="c31">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c39">k</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green" title="E83">A32</font></i></a>, <a class="txt" href="yellow_6.html#E20"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E47:121_4_6"/>
<font color="Maroon" title="c31">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c36">p</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E39:121_4_6"><i><font color="Green" title="E109">A88</font></i></a>, <a class="txt" href="yellow_6.html#E40:121_4_6"><i><font color="Green" title="E110">A89</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E42:121_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow_6.html#E14:121_4"><i><font color="Green" title="E82">A31</font></i></a>, <a class="txt" href="yellow_6.html#E38:121_4"><i><font color="Green" title="E92">A49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:121"/>
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T49" target="_self" title="YELLOW_6:th.49">Th49</a></i>;<br/>
<b>hence </b><a NAME="E10:121"/>
 <a href="yellow_6.html#K13" title="YELLOW_6:func.13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E8:121"><i><font color="Green" title="E75">A24</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
