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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">N</font> be  <a href="yellow_6.html#V1">constant</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font>;<br/>





<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">N</font><b> such that </b><br/><a NAME="E2:26"/><i><font color="Green">E25</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:26"/>
for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">p</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green">E28</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green">E29</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:26"/><i><font color="Green">E30</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:26"/>
for <font color="Olive">n1</font>, <font color="Olive">n2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> &amp; <font color="Olive">n2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> holds <br/>the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">n2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:26_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E2:26_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font>
 ;<br/>

<a NAME="E3:26_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:26_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b>the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> = 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel28.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="waybel28.html#E2:26"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
<b>by </b><i>, <a class="ref" href="waybel28.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:26"/><b>then </b><i><font color="Green">E77</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E9:26"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">M</font> is <a href="yellow_6.html#V1">constant</a>
 <b>by </b><i><a class="ref" href="yellow_6.html#D6">YELLOW_6:def 6</a></i>;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:26"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E12:26"/><i><font color="Green">E80</font></i>: 
 <a href="yellow_6.html#K1">the_value_of</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D1">YELLOW_6:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font>;<br/>
<a NAME="E14:26"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="waybel28.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then </b><i><font color="Green">E81</font></i>:  <a href="yellow_6.html#K1">the_value_of</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> = 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="txt" href="waybel28.html#E2:26"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/>
<b>thus </b> <a href="yellow_6.html#K5">the_value_of</a> <font color="Maroon">N</font> = 
 <a href="yellow_6.html#K1">the_value_of</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="yellow_6.html#D10">YELLOW_6:def 10</a></i>
<br/>.= 
 <a href="yellow_6.html#K1">the_value_of</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font>
<b>by </b><i>, , , <a class="ref" href="yellow_6.html#D1">YELLOW_6:def 1</a></i>
<br/>.= 
 <a href="yellow_6.html#K5">the_value_of</a> <font color="Maroon">M</font>
<b>by </b><i>, <a class="ref" href="yellow_6.html#D10">YELLOW_6:def 10</a></i>
;<br/>


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