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<b>let </b><font color="Maroon">IT1</font> be  <a href="glib_003.html#V15">real-weighted</a> <a href="glib_003.html#V17">real-elabeled</a> <a href="glib_003.html#NM35">WEVGraphSeq</a>, <font color="Maroon">IT2</font> be  <a href="glib_003.html#V15">real-weighted</a> <a href="glib_003.html#V17">real-elabeled</a> <a href="glib_003.html#NM35">WEVGraphSeq</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_1_2"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">IT1</font> <a href="glib_004.html#K5">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<span class="p1">(<span class="default"><font color="Maroon">source</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> &amp; ( 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">IT1</font> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K4" target="_self">AP:Step</a> <span class="p1">(<span class="default"><font color="Maroon">IT1</font> <a href="glib_004.html#K5">.</a> <font color="Olive">n</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:40_1_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">IT2</font> <a href="glib_004.html#K5">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<span class="p1">(<span class="default"><font color="Maroon">source</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> &amp; ( 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">IT2</font> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K4" target="_self">AP:Step</a> <span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="glib_004.html#K5">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">IT1</font> <a href="glib_004.html#K5">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="glib_004.html#K5">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:40_1_2"/>
<font color="Maroon">IT1</font> <a href="glib_004.html#K5">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="glib_004.html#K5">.</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:40_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:40_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:40_1_2_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">IT1</font> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 ;<br/><font color="Maroon">IT1</font> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="glib_005.html#K4" target="_self">AP:Step</a> <span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">IT2</font> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/><a NAME="E4:40_1_2_1"/><b>then </b>
<font color="Maroon">IT1</font> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>hence </b><a NAME="E5:40_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E6:40_1_2"/><b>then </b><i><font color="Green">E112</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E7:40_1_2"/>
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">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E8:40_1_2"/><b>then </b>
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">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 
;<br/>
<b>hence </b><a NAME="E9:40_1_2"/>
<font color="Maroon">IT1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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