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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="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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 <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="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> holds <br/>( ( <font color="Olive" title="b2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b1">X</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> ) &amp; ( <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b2">Y</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</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 <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="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> holds <br/>( ( <font color="Olive" title="b1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">X</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> ) &amp; ( <font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b1">Y</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</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"> ( <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="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> implies ( ( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">X</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> ) &amp; ( <font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">Y</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> ) ) )</span><br/>





<b>assume </b><a NAME="E1:62"/><i><font color="Green" title="E28">A1</font></i>: 
for <font color="Olive" title="b1">D</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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 <font color="Olive" title="b1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> holds <br/> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">D</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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D2" title="WAYBEL11:def.2">WAYBEL11:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">X</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> ) &amp; ( <font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">Y</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> ) )</span><br/>

<div><b>hereby </b> <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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">Y</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a></span><br/><b>thus </b><a NAME="E2:62_1"/>
<font color="Maroon" title="c3">X</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D2" title="WAYBEL11:def.2">WAYBEL11:def 2</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">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> or  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )</span><br/>

<b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:62_1_1"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c6">t</font> <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="#E1:62_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">t'</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">t</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <b>by </b><i><a class="txt" href="#E3:62_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="waybel_0.html#T5" title="WAYBEL_0:th.5">WAYBEL_0:5</a></i>;<br/>
<a NAME="E5:62_1_1"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c7">t'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="txt" href="#E3:62_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:62_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">t'</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span>,<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>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>
<b>then </b><i><font color="Green" title="E33">A6</font></i>:  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">t'</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span> = 
<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#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <span class="p4"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">t'</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p4"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">t'</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="yellow_3.html#T46" title="YELLOW_3:th.46">YELLOW_3:46</a></i>
<br/>.= 
<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#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p4"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">t'</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T11" title="FUNCT_5:th.11">FUNCT_5:11</a></i>
<br/>.= 
<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#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">t'</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T11" title="FUNCT_5:th.11">FUNCT_5:11</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span>,<font color="Maroon" title="c6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E3:62_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="yellow_0.html#T39" title="YELLOW_0:th.39">YELLOW_0:39</a></i>
;<br/>
<a NAME="E8:62_1_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">t'</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</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="#E1:62"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E1:62_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E5:62_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E9:62_1_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font> <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="#E7:62_1_1"><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"> verum</span><br/>


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

<b>assume </b><a NAME="E3:62"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">Y</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a></span><br/>

<b>let </b><font color="Maroon" title="c5">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D2" title="WAYBEL11:def.2">WAYBEL11:def 2</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">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> or  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )</span><br/>

<b>assume </b><a NAME="E4:62"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:62"/><i><font color="Green" title="E31">A9</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>
 <b>by </b><i><a class="txt" href="#E3:62"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">s'</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="#E6:62"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="waybel_0.html#T5" title="WAYBEL_0:th.5">WAYBEL_0:5</a></i>;<br/>
<a NAME="E8:62"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c7">s'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="#E6:62"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:62"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">s'</font>,<font color="Maroon" title="c5">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span>,<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>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>
<b>then </b><i><font color="Green" title="E33">A11</font></i>:  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">s'</font>,<font color="Maroon" title="c5">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span> = 
<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#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <span class="p4"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">s'</font>,<font color="Maroon" title="c5">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p4"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">s'</font>,<font color="Maroon" title="c5">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="yellow_3.html#T46" title="YELLOW_3:th.46">YELLOW_3:46</a></i>
<br/>.= 
<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#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">s'</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p4"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">s'</font>,<font color="Maroon" title="c5">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</a></span></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T11" title="FUNCT_5:th.11">FUNCT_5:11</a></i>
<br/>.= 
<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#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">s'</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="ref" href="funct_5.html#T11" title="FUNCT_5:th.11">FUNCT_5:11</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E6:62"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="yellow_0.html#T39" title="YELLOW_0:th.39">YELLOW_0:39</a></i>
;<br/>
<a NAME="E11:62"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1"><a href="yellow_3.html#K10" title="YELLOW_3:func.10">[:</a><span class="default"><font color="Maroon" title="c7">s'</font>,<font color="Maroon" title="c5">D</font></span><a href="yellow_3.html#K10" title="YELLOW_3:func.10">:]</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="#E1:62"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E4:62"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="#E8:62"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E12:62"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font> <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="#E10:62"><i><font color="Green" title="E33">A11</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"> verum</span><br/>


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