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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>





<a NAME="E1:53"/><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">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</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">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">t</font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">t</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">x1</font>,<font color="Maroon">x2</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>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self">Th4</a></i>;<br/><b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E6:53_1"/>
( <font color="Maroon">s</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">t</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self">Th4</a>, <a class="ref" href="waybel_8.html#T4">WAYBEL_8:4</a></i>;<br/><a NAME="E7:53_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/><a NAME="E8:53_1"/><i><font color="Green">E23</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E9:53_1"/>
( <font color="Maroon">x1</font> is <a href="waybel_3.html#V1">compact</a> &amp; <font color="Maroon">x2</font> is <a href="waybel_3.html#V1">compact</a> )
 <b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self">Th4</a>, <a class="ref" href="waybel_8.html#T4">WAYBEL_8:4</a></i>;<br/><a NAME="E10:53_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="yellow_3.html#K7">]</a></span> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E11:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow10.html#T5" target="_self">Th5</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:53"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span> ;<br/>
<a NAME="E5:53"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E6:53"/><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> is <a href="waybel_3.html#V1">compact</a> )
 
<b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self">Th4</a>, <a class="ref" href="waybel_8.html#T4">WAYBEL_8:4</a></i>;<br/>
<a NAME="E7:53"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">s</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x'</font> <a href="yellow_3.html#K8">`1</a>  &amp; <font color="Maroon">t</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x'</font> <a href="yellow_3.html#K9">`2</a>  )
 
<b>by </b><i><a class="ref" href="yellow10.html#T5" target="_self">Th5</a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E8:53"/>
( <font color="Maroon">x'</font> <a href="yellow_3.html#K8">`1</a>  is <a href="waybel_3.html#V1">compact</a> &amp; <font color="Maroon">x'</font> <a href="yellow_3.html#K9">`2</a>  is <a href="waybel_3.html#V1">compact</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:53"/><b>then </b>
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">t</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:53"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">t</font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow10.html#T5" target="_self">Th5</a>, <a class="ref" href="mcart_1.html#T11">MCART_1:11</a></i>;<br/>


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