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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<a NAME="E1:9"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:9_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="waybel_9.html#D7">WAYBEL_9:def 7</a></i>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E7:9_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="waybel_9.html#T16">WAYBEL_9:16</a></i>
;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><a NAME="E3:9"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <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">j</font> )
 ;<br/>

<a NAME="E4:9"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_9.html#D7">WAYBEL_9:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i>, <a class="ref" href="waybel_9.html#D7">WAYBEL_9:def 7</a></i>;<br/>
<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="waybel_9.html#T16">WAYBEL_9:16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>

;<br/>
<b>hence </b><a NAME="E7:9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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