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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="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"> (  <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> #) &amp; <font color="Maroon" title="c1">L1</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> implies <font color="Maroon" title="c2">L2</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="E1:30"/><i><font color="Green" title="E19">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="E2:30"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c1">L1</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</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">L2</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a></span><br/>

<a NAME="E3:30"/><i><font color="Green" title="E21">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> ) ) &amp; <font color="Maroon" title="c1">L1</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">L1</font> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a> )
 
<b>by </b><i><a class="txt" href="waybel_8.html#E2:30"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self" title="WAYBEL_8:def.4">Def4</a></i>;<br/>
<a NAME="E4:30"/><b>then </b><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c2">L2</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E1:30"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="waybel_8.html#T15" target="_self" title="WAYBEL_8:th.15">Th15</a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green" title="E23">A5</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">L2</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </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>; <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">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="c3">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )</span><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#E1:30"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<a NAME="E2:30_1"/><i><font color="Green" title="E24">A6</font></i>: 
( 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#E2:30"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self" title="WAYBEL_8:def.4">Def4</a></i>;<br/>
<a NAME="E3:30_1"/>
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c4">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="c3">x</font>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E1:30"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="waybel_8.html#E3:30"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="waybel_8.html#E4:30"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="waybel_8.html#T10" target="_self" title="WAYBEL_8:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1"/>
( not  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c3">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="c3">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#E1:30"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="waybel_8.html#E2:30_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="waybel_0.html#T3" title="WAYBEL_0:th.3">WAYBEL_0:3</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>
<a NAME="E6:30"/>
<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="txt" href="waybel_8.html#E1:30"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="waybel_8.html#E3:30"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="waybel_8.html#E4:30"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="waybel_8.html#T16" target="_self" title="WAYBEL_8:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E7:30"/>
<font color="Maroon" title="c2">L2</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:30"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="waybel_8.html#E5:30"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self" title="WAYBEL_8:def.4">Def4</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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