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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="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</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"> ( <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> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> implies ( <font color="Maroon" title="c1">S</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green" title="E30">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 <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 <br/> ( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Olive" title="b1">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green" title="E31">A2</font></i>: 
( <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> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <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> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> )</span><br/>

<a NAME="E3:82"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c1">S</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> )
 
<b>by </b><i><a class="txt" href="yellow10.html#E2:82"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="waybel_2.html#T11" title="WAYBEL_2:th.11">WAYBEL_2:11</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">T</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a></span>
<div class="add"><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="c1">S</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">S</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> ( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">s</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">s</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )</span><br/><b>consider </b><font color="Maroon" title="c4">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>;<br/><a NAME="E2:82_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
<span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">t</font></span>)</span></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></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"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow10.html#E3:82"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="yellow10.html#T50" target="_self" title="YELLOW10:th.50">Th50</a></i>;<br/><a NAME="E3:82_1_1"/><i><font color="Green" title="E34">A5</font></i>: 
 <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">t</font></span>)</span></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="ref" href="funct_5.html#T11" title="FUNCT_5:th.11">FUNCT_5:11</a></i>;<br/><a NAME="E4:82_1_1"/>
 <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"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 <b>by </b><i><a class="txt" href="yellow10.html#E1:82"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:82_1_1"/>
( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">s</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">s</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 <b>by </b><i><a class="txt" href="yellow10.html#E2:82_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="yellow10.html#E3:82_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="yellow_3.html#T22" title="YELLOW_3:th.22">YELLOW_3:22</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><b>thus </b><a NAME="E2:82_1"/>
<font color="Maroon" title="c1">S</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 <b>by </b><i><a class="txt" href="yellow10.html#E2:82"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="waybel_2.html#T11" title="WAYBEL_2:th.11">WAYBEL_2:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a></span><br/><b>thus </b><a NAME="E3:82_1"/>
<font color="Maroon" title="c1">S</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</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="82_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</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">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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">s</font></span>)</span>,<font color="Maroon" title="c1">S</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>;<br/>
<a NAME="E2:82_1_2"/>
( not  <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"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <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"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<b>by </b><i><a class="txt" href="yellow10.html#E1:82"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E3:82_1_2"/><b>then </b>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</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"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="txt" href="yellow10.html#E2:82"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E4:82_1_2"/><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">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span> <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"><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">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p5"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</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">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p5"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</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="yellow10.html#T5" target="_self" title="YELLOW10:th.5">Th5</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">s</font> = 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</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> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E2:82"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow10.html#E4:82_1_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">t</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> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow10.html#E3:82"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="yellow10.html#T52" target="_self" title="YELLOW10:th.52">Th52</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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">s</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</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>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">T</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c3">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> ( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">t</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )</span><br/><b>consider </b><font color="Maroon" title="c4">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><a NAME="E2:82_2"/><i><font color="Green" title="E33">A7</font></i>: 
<span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">t</font></span>)</span></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></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"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow10.html#E3:82"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="yellow10.html#T50" target="_self" title="YELLOW10:th.50">Th50</a></i>;<br/><a NAME="E3:82_2"/><i><font color="Green" title="E34">A8</font></i>: 
 <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">t</font></span>)</span></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">t</font>
 <b>by </b><i><a class="ref" href="funct_5.html#T11" title="FUNCT_5:th.11">FUNCT_5:11</a></i>;<br/><a NAME="E4:82_2"/>
 <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"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 <b>by </b><i><a class="txt" href="yellow10.html#E1:82"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:82_2"/>
( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">t</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">t</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 <b>by </b><i><a class="txt" href="yellow10.html#E2:82_2"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="yellow10.html#E3:82_2"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="yellow_3.html#T22" title="YELLOW_3:th.22">YELLOW_3:22</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>

<b>thus </b><a NAME="E6:82"/>
<font color="Maroon" title="c2">T</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 <b>by </b><i><a class="txt" href="yellow10.html#E2:82"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="waybel_2.html#T11" title="WAYBEL_2:th.11">WAYBEL_2:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">T</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a></span><br/>

<b>let </b><font color="Maroon" title="c3">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</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">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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">t</font></span>)</span>,<font color="Maroon" title="c2">T</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>
<a NAME="E8:82"/>
( not  <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"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <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"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<b>by </b><i><a class="txt" href="yellow10.html#E1:82"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E9:82"/><b>then </b>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</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"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="txt" href="yellow10.html#E2:82"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E10:82"/><b>then </b><i><font color="Green" title="E33">A9</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span> <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"><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">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p5"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</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">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p5"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</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="yellow10.html#T5" target="_self" title="YELLOW10:th.5">Th5</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">t</font> = 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</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> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E2:82"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p2">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow10.html#E10:82"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">t</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> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow10.html#E3:82"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="yellow10.html#T53" target="_self" title="YELLOW10:th.53">Th53</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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">t</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</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>
