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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E19</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #)
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V2" target="_self">algebraic</a>
 ;<br/>

<a NAME="E3:30"/><i><font color="Green">E21</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> ) ) &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V1" target="_self">satisfying_axiom_K</a> )
 
<b>by </b><i><a class="txt" href="waybel_8.html#E2:30"><i><font color="Green">E20</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:30"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V24">up-complete</a>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E1:30"><i><font color="Green">E19</font></i></a>, <a class="ref" href="waybel_8.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">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">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel_8.html#E1:30"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E2:30_1"/><i><font color="Green">E24</font></i>: 
( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="txt" href="waybel_8.html#E2:30"><i><font color="Green">E20</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:30_1"/>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E1:30"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel_8.html#E3:30"><i><font color="Green">E21</font></i></a>, <a class="txt" href="waybel_8.html#E4:30"><i><font color="Green">E22</font></i></a>, <a class="ref" href="waybel_8.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1"/>
( not  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i><a class="txt" href="waybel_8.html#E1:30"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel_8.html#E2:30_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel_0.html#T3">WAYBEL_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:30"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_8.html#V1" target="_self">satisfying_axiom_K</a>
 
<b>by </b><i><a class="txt" href="waybel_8.html#E1:30"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel_8.html#E3:30"><i><font color="Green">E21</font></i></a>, <a class="txt" href="waybel_8.html#E4:30"><i><font color="Green">E22</font></i></a>, <a class="ref" href="waybel_8.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E7:30"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_8.html#V2" target="_self">algebraic</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:30"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel_8.html#E5:30"><i><font color="Green">E23</font></i></a>, <a class="ref" href="waybel_8.html#D4" target="_self">Def4</a></i>;<br/>


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