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

<a NAME="E1:78_1_1"/><i><font color="Green" title="E41">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_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">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/>
<b>thus </b><a NAME="E2:78_1_1"/>
 <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c3">J</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">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="c3">J</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5" target="_self" title="YELLOW_6:def.5">YELLOW_6:def 5</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <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="c4">x</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c2">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> 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="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <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="c5">y</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c1">T</font></span>)</span>,<font color="Maroon" title="c2">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> 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="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c1">T</font></span>)</span> );<br/>
<div><i><font color="Green" title="E42">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font>;<br/><b>consider </b><font color="Maroon" title="c7">zi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c6">i</font></span>)</span><b> such that </b><br/><a NAME="E2:78_1_1_1_1"/><i><font color="Green" title="E43">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c6">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">zi</font> &amp; <font color="Maroon" title="c5">y</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c6">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">zi</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5" target="_self" title="YELLOW_6:def.5">Def5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">zi</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">z</font>;<br/><b>thus </b><a NAME="E4:78_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">z</font>]
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:78_1_1_1_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">z</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="c2">Y</font><b> such that </b><br/><a NAME="E3:78_1_1_1"/><i><font color="Green" title="E43">A4</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="c2">Y</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="msualg_9.html#S1" title="MSUALG_9:sch.1">MSUALG_9:sch 1</a>(<a class="txt" href="yellow_6.html#E1:78_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>);<br/></i>
<i><font color="Green" title="E44">A5</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">z</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c3">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:78_1_1_1_3"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c3">J</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c7">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:78_1_1_1_3"/>
<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="c4">x</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c8">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</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> 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="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:78_1_1_1"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/><a NAME="E4:78_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</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="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E5:78_1_1_1_3"/>
<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</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="c3">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self" title="YELLOW_6:th.9">Th9</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">z</font> = <font color="Maroon" title="c6">z</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="c3">J</font></span>)</span> <b>by </b><i><a class="txt" href="yellow_6.html#E1:78_1_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="yellow_6.html#E4:78_1_1_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">z</font>
;<br/>

<a NAME="E7:78_1_1_1"/>
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="c2">Y</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:78_1_1_1_4"/>
<font color="Maroon" title="c8">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="c2">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c8">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">xi</font> = <font color="Maroon" title="c4">x</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c11">zi</font> = <font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c9">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c9">j</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c9">j</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c10">xi</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c11">zi</font>
;<br/>

<b>thus </b><a NAME="E4:78_1_1_1_4"/>
( <font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c10">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c11">zi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )
 ;<br/>

<a NAME="E5:78_1_1_1_4"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">xi</font>,<font color="Maroon" title="c11">zi</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E3:78_1_1_1"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:78_1_1_1_4"/>
<font color="Maroon" title="c10">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">zi</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:78_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:78_1_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>

<a NAME="E9:78_1_1_1"/>
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="c2">Y</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">yi</font>, <font color="Olive" title="b4">zi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">zi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">yi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">zi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:78_1_1_1_5"/>
<font color="Maroon" title="c8">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="c2">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c8">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">yi</font> = <font color="Maroon" title="c5">y</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c11">zi</font> = <font color="Maroon" title="c7">z</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c9">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c9">j</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c9">j</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c10">yi</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c11">zi</font>
;<br/>

<b>thus </b><a NAME="E4:78_1_1_1_5"/>
( <font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c10">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c11">zi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )
 ;<br/>

<a NAME="E5:78_1_1_1_5"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">yi</font>,<font color="Maroon" title="c11">zi</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E3:78_1_1_1"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:78_1_1_1_5"/>
<font color="Maroon" title="c10">yi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">zi</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:78_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:78_1_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</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="c3">J</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:78_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font>
 <b>and </b><br/><a NAME="E4:78_1_1"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">z</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">fx</font>, <font color="Maroon" title="c8">gy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:78_1_1"/><i><font color="Green" title="E44">A8</font></i>: 
( <font color="Maroon" title="c7">fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c8">gy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <b>and </b><br/><a NAME="E7:78_1_1"/><i><font color="Green" title="E45">A9</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="c2">Y</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">gy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:78_1_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="yellow_6.html#E3:78_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">fy</font>, <font color="Maroon" title="c10">gz</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:78_1_1"/><i><font color="Green" title="E46">A10</font></i>: 
( <font color="Maroon" title="c9">fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c10">gz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> )
 <b>and </b><br/><a NAME="E10:78_1_1"/><i><font color="Green" title="E47">A11</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="c2">Y</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">gz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:78_1_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="yellow_6.html#E4:78_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E11:78_1_1"/>
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="c2">Y</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_2"><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> ;<br/>

<b>assume </b><a NAME="E1:78_1_1_2"/><i><font color="Green" title="E48">A12</font></i>: 
<font color="Maroon" title="c11">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="c2">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">j</font> = <font color="Maroon" title="c11">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<b>consider </b><font color="Maroon" title="c13">R1</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> , <font color="Maroon" title="c14">xi</font>, <font color="Maroon" title="c15">yi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">R1</font><b> such that </b><br/><a NAME="E4:78_1_1_2"/><i><font color="Green" title="E49">A13</font></i>: 
( <font color="Maroon" title="c13">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c14">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c15">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c14">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">yi</font> )
 <b>by </b><i><a class="txt" href="yellow_6.html#E6:78_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="yellow_6.html#E7:78_1_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="yellow_6.html#E1:78_1_1_2"><i><font color="Green" title="E48">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">R2</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> , <font color="Maroon" title="c17">yi'</font>, <font color="Maroon" title="c18">zi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">R2</font><b> such that </b><br/><a NAME="E6:78_1_1_2"/><i><font color="Green" title="E50">A14</font></i>: 
( <font color="Maroon" title="c16">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c17">yi'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c18">zi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c17">yi'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">zi</font> )
 <b>by </b><i><a class="txt" href="yellow_6.html#E9:78_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="yellow_6.html#E10:78_1_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="yellow_6.html#E1:78_1_1_2"><i><font color="Green" title="E48">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">xi</font> = <font color="Maroon" title="c14">xi</font>, <font color="Maroon" title="c20">zi</font> = <font color="Maroon" title="c18">zi</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <b>by </b><i><a class="txt" href="yellow_6.html#E4:78_1_1_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="yellow_6.html#E6:78_1_1_2"><i><font color="Green" title="E50">A14</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c12">j</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c19">xi</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c20">zi</font>
;<br/>

<b>thus </b><a NAME="E8:78_1_1_2"/>
( <font color="Maroon" title="c3">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c19">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c20">zi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> )
 <b>by </b><i><a class="txt" href="yellow_6.html#E4:78_1_1_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="yellow_6.html#E6:78_1_1_2"><i><font color="Green" title="E50">A14</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:78_1_1_2"/>
<font color="Maroon" title="c19">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">zi</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E4:78_1_1_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="yellow_6.html#E6:78_1_1_2"><i><font color="Green" title="E50">A14</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="E12:78_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:78_1_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>


</div>
