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

<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">S2</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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font>;<br/>





<b>set </b><font color="Maroon">I1</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S1</font>;<br/>
<b>set </b><font color="Maroon">I2</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S2</font>;<br/>
<b>set </b><font color="Maroon">x</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E1:28"/><i><font color="Green">E18</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p3"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p3"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p3"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p3"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:28"/><i><font color="Green">E20</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>thus </b><a NAME="E3:28"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d</font> implies <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 ;<br/>

<a NAME="E2:28_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p5"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p5"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></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 <font color="Maroon">S1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p5"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p5"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></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 <font color="Maroon">S2</font> )
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E3:28_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K2" target="_self">["</a><span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S2</font></span><a href="yellow_3.html#K2" target="_self">"]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></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"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a><br/>


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

<b>assume </b><a NAME="E4:28"/>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 ;<br/>

<a NAME="E5:28"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></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"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E6:28"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>,<span class="p2"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K2" target="_self">["</a><span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S2</font></span><a href="yellow_3.html#K2" target="_self">"]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:28"/>
( <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S1</font> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S2</font> )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a><br/>


</div>
