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

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> holds   <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> holds   <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><br/>





<a NAME="E1:56"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E2:56"/><b>then </b><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>thus </b><a NAME="E3:56"/>
 <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="yellow10.html#T51" target="_self" title="YELLOW10:th.51">Th51</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">x</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">x</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E4:56"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">x</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<a NAME="E6:56"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c5">a'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  &amp; <font color="Maroon" title="c5">a'</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 
<b>by </b><i><a class="txt" href="yellow10.html#E4:56"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/>
<a NAME="E7:56"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/>
<a NAME="E8:56"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c5">a'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow10.html#E6:56"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>
<a NAME="E9:56"/>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c5">a'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c5">a'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E10:56"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c5">a'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="yellow10.html#E6:56"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="yellow10.html#T23" target="_self" title="YELLOW10:th.23">Th23</a>, <a class="ref" href="waybel_3.html#T15" title="WAYBEL_3:th.15">WAYBEL_3:15</a></i>;<br/>
<a NAME="E11:56"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c5">a'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow10.html#E8:56"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:56"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow10.html#E2:56"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
