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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/><font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font>;<br/>
<b>consider </b><font color="Maroon">R</font> being    <a href="relset_1.html#M2">Relation</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:18_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 <b>from </b><i><a class="ref" href="relset_1.html#S2">RELSET_1:sch 2</a>();<br/></i>
<b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="relset_1.html#NM3">Relation</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>take </b><font color="Maroon">N</font> =  <a href="waybel_0.html#G1">NetStr</a>(# <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">R</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> </span>)</span> #);<br/>

<b>thus </b><a NAME="E4:18_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>thus </b><a NAME="E5:18_1_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> 
 ;<br/>

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">i'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E6:18_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j'</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">i</font>, <font color="Maroon">y</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E8:18_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:18_1_1"/>
( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> iff <font color="Maroon">i'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j'</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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