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<b>let </b><font color="Maroon" title="c1">S</font>, <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#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">X</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> &amp; <font color="Olive" title="b2">Y</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> holds <br/><span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a></span><br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>; <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">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Maroon" title="c3">X</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> &amp; <font color="Olive" title="b1">Y</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> holds <br/><span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a></span><br/><b>let </b><font color="Maroon" title="c4">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> &amp; <font color="Maroon" title="c4">Y</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> implies <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green" title="E28">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> )
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> holds <br/> ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b2">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D1" title="WAYBEL_6:def.1">WAYBEL_6:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a></span><br/>

<b>let </b><font color="Maroon" title="c5">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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D1" title="WAYBEL_6:def.1">WAYBEL_6:def 1</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> or  ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(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>) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c5">x</font> ) )</span><br/>

<b>assume </b><a NAME="E3:59"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(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>) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<a NAME="E4:59"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c5">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="c5">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="c5">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/>
<a NAME="E5:59"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="yellow10.html#E3:59"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E7:59"/><i><font color="Green" title="E32">A5</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">s</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  )
 <b>by </b><i><a class="txt" href="yellow10.html#E1:59"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E8:59"/>
<font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="txt" href="yellow10.html#E3:59"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="yellow10.html#E4:59"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font><b> such that </b><br/><a NAME="E10:59"/><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c7">t</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  )
 <b>by </b><i><a class="txt" href="yellow10.html#E2:59"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c7">t</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<b>take </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">t</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> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<b>thus </b><a NAME="E12:59"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">t</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> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow10.html#E7:59"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="yellow10.html#E10:59"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c5">x</font></span><br/>

<b>thus </b><a NAME="E13:59"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="yellow10.html#E4:59"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="yellow10.html#E7:59"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="yellow10.html#E10:59"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="yellow10.html#T19" target="_self" title="YELLOW10:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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