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

<b>let </b><font color="Maroon">G1</font> be    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>, <font color="Maroon">G2</font> be    <a href="measure7.html#M1" target="_self">Interval_Covering</a> of  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">G1</font> <a href="measure7.html#K3" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E2:31_1_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">G2</font> <a href="measure7.html#K3" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>

<a NAME="E3:31_1_2"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:31_1_2_1"/>
<font color="Maroon">G1</font> <a href="measure7.html#K3" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="measure7.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" target="_self">pr2</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:31_1_2_1"/>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1_2"/>
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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