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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <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">L</font>, <font color="Maroon">g</font> be   <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">L</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">L</font> <a href="yellow_1.html#K6">|^</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">L</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:26"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 ;<br/>

<b>set </b><font color="Maroon">J</font> = <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">L</font>;<br/>
<a NAME="E2:26"/><i><font color="Green">E32</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">L</font></span>)</span></span>)</span> &amp; <font color="Maroon">L</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a>, <a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E3:26"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> iff  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">x</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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">L</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> ) ) ) )
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><b>thus </b><a NAME="E2:26_1"/>
<font color="Maroon">f</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a><br/>

<b>assume </b><a NAME="E1:26_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:26_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> &amp;  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">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">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="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">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="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:26_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</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="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> )
 ;<br/>


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

<b>assume </b><a NAME="E5:26"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font>, <font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><b>take </b><font color="Maroon">f'</font> = <font color="Maroon">f'</font>;<br/><b>take </b><font color="Maroon">g'</font> = <font color="Maroon">g'</font>;<br/><b>thus </b><a NAME="E2:26_2"/>
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:26_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E4:26_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</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="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> ) )
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E5:26_2"/>
 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">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">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="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:26"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>


</div>
