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<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green" title="E18">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">L2</font> #)
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c1">L1</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a>
 <b>and </b><br/><a NAME="E3:29"/><i><font color="Green" title="E20">A3</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">L1</font> 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> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><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 <font color="Maroon" title="c2">L2</font>;<br/><b>reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="waybel_8.html#E1:29"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/><a NAME="E2:29_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c4">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">x'</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:29"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="waybel_8.html#D3" target="_self" title="WAYBEL_8:def.3">Def3</a></i>;<br/><a NAME="E3:29_1"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">x</font> <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="c4">x'</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:29"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="waybel_8.html#T10" target="_self" title="WAYBEL_8:th.10">Th10</a></i>;<br/><a NAME="E4:29_1"/>
( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">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="Maroon" title="c4">x'</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:29"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/><a NAME="E5:29_1"/><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> <font color="Maroon" title="c4">x'</font>,<font color="Maroon" title="c1">L1</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/><b>hence </b><a NAME="E6:29_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:29"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="waybel_8.html#E2:29_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="waybel_8.html#E3:29_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="yellow_0.html#T26" title="YELLOW_0:th.26">YELLOW_0:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:29"/>
<font color="Maroon" title="c2">L2</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#D3" target="_self" title="WAYBEL_8:def.3">Def3</a></i>;<br/>


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