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

<b>set </b><font color="Maroon">IT</font> = <font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">xa</font>, <font color="Olive">ya</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">ya</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">xa</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">ya</font> );<br/>
<a NAME="E3:57_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:57_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">x'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>, <font color="Maroon">ya</font> = <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> = <font color="Maroon">xa</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">ya</font>;<br/>

<b>thus </b><a NAME="E3:57_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 ;<br/>

<b>take </b>
<font color="Maroon">xa</font>
;<br/>

<b>take </b>
<font color="Maroon">ya</font>
;<br/>

<b>thus </b><a NAME="E4:57_1_1_1"/>
( <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">ya</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">ya</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:57_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E6:57_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E8:57_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span> ;<br/>
<a NAME="E2:57_1_1_2"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:57_1_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:57_1_1_2"/>
 ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>take </b>
<font color="Maroon">x'</font>
;<br/>

<b>thus </b><a NAME="E1:57_1_1_2_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:57_1_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">R</font> = <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K4" target="_self">.</a> <font color="Maroon">i</font> as    <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<a NAME="E5:57_1_1_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xi</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>, <font color="Maroon">yi</font> = <font color="Maroon">x'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">R</font>
;<br/>

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

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

<b>consider </b><font color="Maroon">xa</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">ya</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E8:57_1_1_2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">ya</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">ya</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E9:57_1_1_2_1"/>
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yi</font> )
 <b>by </b><i>, <a class="txt" href="yellow_1.html#E5:57_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:57_1_1_2"/><b>then </b>
<font color="Maroon">z1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:57_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E7:57_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:57_1_1_2"/>
<font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:57_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font>, <font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span> ;<br/>
<a NAME="E2:57_1_1_3"/>
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:57_1_1_3"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:57_1_1_3"/>
 ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_3_1"><b>proof </b></a><div class="add">

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

<b>take </b>
<font color="Maroon">y'</font>
;<br/>

<b>thus </b><a NAME="E1:57_1_1_3_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:57_1_1_3_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">R</font> = <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K4" target="_self">.</a> <font color="Maroon">i</font> as    <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<a NAME="E5:57_1_1_3_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xi</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>, <font color="Maroon">yi</font> = <font color="Maroon">y'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">R</font>
;<br/>

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

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

<b>consider </b><font color="Maroon">xa</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">ya</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E8:57_1_1_3_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">ya</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">ya</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E9:57_1_1_3_1"/>
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yi</font> )
 <b>by </b><i><a class="txt" href="yellow_1.html#E5:57_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow_1.html#E8:57_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:57_1_1_3"/><b>then </b>
<font color="Maroon">z1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:57_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E7:57_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:57_1_1_3"/>
<font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:57_1_1"/>
for <font color="Olive">z'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Olive">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> holds <br/><font color="Olive">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:57_1_1_4"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:57_1_1_4"/>
<font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_4_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">z'</font>, <font color="Maroon">z3</font> = <font color="Maroon">z</font>, <font color="Maroon">z4</font> = <font color="Maroon">y</font>, <font color="Maroon">z5</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">z2</font>, <font color="Maroon">K</font> = <font color="Maroon">z3</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E3:57_1_1_4_1"/>
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:57_1_1_4_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:57_1_1_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z'</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E6:57_1_1_4_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">z5</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E7:57_1_1_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z'</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E8:57_1_1_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">z4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E9:57_1_1_4_1"/><b>then </b><i><font color="Green">E120</font></i>: 
( <font color="Maroon">z2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z5</font> &amp; <font color="Maroon">z2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z4</font> )
 
<b>by </b><i><a class="txt" href="yellow_1.html#E8:57_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:57_1_1_4_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z5</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> ) ) )
 <b>by </b><i><a class="txt" href="yellow_1.html#E5:57_1_1"><i><font color="Green">E24</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:57_1_1_4_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z4</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> ) ) )
 <b>by </b><i><a class="txt" href="yellow_1.html#E5:57_1_1"><i><font color="Green">E24</font></i></a>, , </i>;<br/>
<a NAME="E14:57_1_1_4_1"/>
 ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z3</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">J</font>
;<br/>

<b>take </b>
<font color="Maroon">K</font>
;<br/>

<b>thus </b><a NAME="E1:57_1_1_4_1_1"/>
( <font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z3</font> )
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:57_1_1_4_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">R</font> = <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K4" target="_self">.</a> <font color="Maroon">i</font> as    <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<a NAME="E5:57_1_1_4_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xi</font> = <font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>, <font color="Maroon">yi</font> = <font color="Maroon">K</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">R</font>
;<br/>

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

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

<b>consider </b><font color="Maroon">xa</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">ya</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E8:57_1_1_4_1_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">ya</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">ya</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R1</font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">xi1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font>, <font color="Maroon">yi1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font><b> such that </b><br/><a NAME="E10:57_1_1_4_1_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K4" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">yi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yi1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R2</font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">xi2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>, <font color="Maroon">yi2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font><b> such that </b><br/><a NAME="E12:57_1_1_4_1_1"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K4" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">yi2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yi2</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E13:57_1_1_4_1_1"/>
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yi</font> )
 <b>by </b><i>, , , , , , <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:57_1_1_4_1"/><b>then </b>
<font color="Maroon">z2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z3</font>
 
<b>by </b><i><a class="txt" href="yellow_1.html#E5:57_1_1"><i><font color="Green">E24</font></i></a>, </i>;<br/>
<a NAME="E16:57_1_1_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">z3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E17:57_1_1_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">z3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E18:57_1_1_4_1"/>
<font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:57_1_1_4"/>
<font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:57_1_1"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_1.html#K6" target="_self">|^</a> <font color="Maroon">X</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> ) ) )
 <b>by </b><i><a class="txt" href="yellow_1.html#E3:57_1_1"><i><font color="Green">E21</font></i></a>, </i>;<br/>


</div>
