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<b>let </b><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font> be    <a href="measure7.html#M1" title="MEASURE7:mode.1">Interval_Covering</a> of  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">F</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_2"/><i><font color="Green" title="E27">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">G1</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E2:31_1_2"/><i><font color="Green" title="E28">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">G2</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 ;<br/>

<a NAME="E3:31_1_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c5">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c7">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:31_1_2_1"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c7">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:31_1_2_1"/>
<font color="Maroon" title="c5">G1</font> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="measure7.html#K5" title="MEASURE7:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" title="MEASURE7:func.12">pr1</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="measure7.html#K3" title="MEASURE7:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure7.html#K13" title="MEASURE7:func.13">pr2</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure7.html#E1:31_1_2"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:31_1_2_1"/>
<font color="Maroon" title="c5">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="measure7.html#E2:31_1_2"><i><font color="Green" title="E28">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1_2"/>
<font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>


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