<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D9" title="WAYBEL11:def.9">WAYBEL11:def 9</a><br/>

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
(  <a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="waybel32.html#K1" title="WAYBEL32:func.1">*'</a> <font color="Maroon" title="c5">f</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon" title="c5">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Olive" title="b2">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> </span>}</span> ,<font color="Maroon" title="c1">R</font> ) )
 <b>by </b><i><a class="ref" href="waybel32.html#D4" target="_self" title="WAYBEL32:def.4">Def4</a></i>;<br/>
<a NAME="E4:41_1_1"/><i><font color="Green" title="E21">A3</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 <span class="p1">(<span class="default"><a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font></span>)</span> #) <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">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">N</font> #)
 
<b>by </b><i><a class="txt" href="waybel32.html#E3:41_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="waybel32.html#D3" target="_self" title="WAYBEL32:def.3">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">i'</font> = <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c7">j'</font> = <font color="Maroon" title="c4">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> = <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c1">R</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>] <b>means </b><font color="Maroon" title="c1">R</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">j'</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>] <b>means </b><font color="Maroon" title="c1">R</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">i'</font>;<br/>
<b>set </b><font color="Maroon" title="c8">J</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c9">I</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> ;<br/>
<a NAME="E6:41_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c10">j0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E8:41_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c10">j0</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">j'</font> &amp; <font color="Maroon" title="c10">j0</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">j'</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5" title="YELLOW_6:def.5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E9:41_1_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel32.html#E8:41_1_1"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">J'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="waybel32.html#E6:41_1_1"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>
<a NAME="E11:41_1_1"/><i><font color="Green" title="E24">A6</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c12">j1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E13:41_1_1"/><i><font color="Green" title="E25">A7</font></i>: 
( <font color="Maroon" title="c12">j1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">i'</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">i'</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5" title="YELLOW_6:def.5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E14:41_1_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel32.html#E13:41_1_1"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">I'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="waybel32.html#E11:41_1_1"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<a NAME="E16:41_1_1"/><i><font color="Green" title="E26">A8</font></i>: 
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c11">J'</font>,<font color="Maroon" title="c1">R</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c13">I'</font>,<font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T76" title="WAYBEL_0:th.76">WAYBEL_0:76</a></i>;<br/>
<a NAME="E17:41_1_1"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c11">J'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">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" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_1_1"/><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">J'</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>consider </b><font color="Maroon" title="c16">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E4:41_1_1_1"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">k</font>
 <b>and </b><br/><a NAME="E5:41_1_1_1"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">j'</font>
 <b>by </b><i><a class="txt" href="waybel32.html#E1:41_1_1_1"><i><font color="Green" title="E28">A10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">i0</font> = <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c18">j0</font> = <font color="Maroon" title="c4">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c19">k'</font> = <font color="Maroon" title="c16">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> ;<br/>
<a NAME="E8:41_1_1_1"/>
( <font color="Maroon" title="c6">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">j'</font> &amp; <font color="Maroon" title="c7">j'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">k'</font> )
 
<b>by </b><i><a class="txt" href="waybel32.html#E1:41_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="waybel32.html#E4:41_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="waybel32.html#E5:41_1_1_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<a NAME="E9:41_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">k'</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:41_1_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">I'</font>
 <b>by </b><i><a class="txt" href="waybel32.html#E4:41_1_1_1"><i><font color="Green" title="E29">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:41_1_1"/><i><font color="Green" title="E28">A13</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c6">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> ,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c5">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c7">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> ,<font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="txt" href="waybel32.html#E3:41_1_1"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>
<a NAME="E19:41_1_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E3:41_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="waybel32.html#D3" target="_self" title="WAYBEL32:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E20:41_1_1"/>
<span class="p1">(<span class="default"><a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="waybel32.html#E16:41_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="waybel32.html#E17:41_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="waybel32.html#E18:41_1_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="yellow_0.html#T35" title="YELLOW_0:th.35">YELLOW_0:35</a></i>;<br/>


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