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<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">x</font> be   <a href="struct_0.html#NM1">Element</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>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</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>;<br/>



<a NAME="E1:29"/><i><font color="Green">E18</font></i>: 
the <a href="struct_0.html#U1">carrier</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:29"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:29"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9" target="_self">`2</a> </span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span> )
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T8">MCART_1:8</a></i>;<br/>
<b>hence </b><a NAME="E7:29"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> iff ( <font color="Maroon">x</font> <a href="yellow_3.html#K8" target="_self">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K8" target="_self">`1</a>  &amp; <font color="Maroon">x</font> <a href="yellow_3.html#K9" target="_self">`2</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K9" target="_self">`2</a>  ) )
 <b>by </b><i/>;<br/>


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