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<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span>, <font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D9">WAYBEL11:def 9</a><br/>

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">N</font>,<font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green">E36</font></i>: 
(  <a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel32.html#K1" target="_self">*'</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> holds  <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">i</font> </span>}</span> ,<font color="Maroon">R</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E4:41_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span> #) <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">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> #)
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> = <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i'</font>;<br/>
<b>set </b><font color="Maroon">J</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">I</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span> ;<br/>
<a NAME="E6:41_1_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>consider </b><font color="Maroon">j0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E8:41_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">j0</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font> &amp; <font color="Maroon">j0</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E9:41_1_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">J'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E11:41_1_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>consider </b><font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E13:41_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">j1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">j1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i'</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E14:41_1_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">I'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E16:41_1_1"/><i><font color="Green">E50</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">J'</font>,<font color="Maroon">R</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">I'</font>,<font color="Maroon">R</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T76">WAYBEL_0:76</a></i>;<br/>
<a NAME="E17:41_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">J'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J'</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:41_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E5:41_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">i0</font> = <font color="Maroon">i</font>, <font color="Maroon">j0</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<a NAME="E8:41_1_1_1"/>
( <font color="Maroon">i'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j'</font> &amp; <font color="Maroon">j'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k'</font> )
 
<b>by </b><i>, , , <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<a NAME="E9:41_1_1_1"/><b>then </b>
<font color="Maroon">i'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k'</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:41_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I'</font>
 <b>by </b><i><a class="ref" href="waybel32.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:41_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span> ,<font color="Maroon">R</font> &amp; <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span> ,<font color="Maroon">R</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E19:41_1_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:41_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i'</font> &amp; <span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j'</font> )
 
;<br/>
<b>hence </b><a NAME="E21:41_1_1"/>
<span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel32.html#K2" target="_self">inf_net</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="waybel32.html#D1" target="_self">Def1</a>, , , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>


</div>
