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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>2</sub></font> is  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Olive">b<sub>1</sub></font> ) ) or ( <font color="Maroon">a<sub>2</sub></font> is  not  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> is  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a> or  not <font color="Maroon">c<sub>3</sub></font> is  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a> ;<br/><a NAME="E3:56_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>4</sub></font>]
 ;<br/><b>hence </b><a NAME="E4:56_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> is  not  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:56_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:56_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
;<br/>
<a NAME="E3:56_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>] holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font>
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:56_1_1"/><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K11" target="_self">PRIM:Init</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S1">RECDEF_1:sch 1</a>(<a class="txt" href="glib_004.html#E2:56_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="glib_004.html#E3:56_1_1"><i><font color="Green">E38</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_004.html#E5:56_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a>;<br/>
<a NAME="E7:56_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="glib_004.html#E5:56_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:56_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a><b> such that </b><br/><a NAME="E3:56_1_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E5:56_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><b>thus </b><a NAME="E4:56_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_004.html#E3:56_1_1_2"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:56_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="glib_004.html#E7:56_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="glib_004.html#E8:56_1_1"><i><font color="Green">E41</font></i></a>);<br/></i>
<a NAME="E10:56_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="glib_000.html#NM2">_Graph</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="glib_000.html#NM11">GraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D55">GLIB_000:def 55</a></i>;<br/>
<a NAME="E12:56_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<sub>1</sub></font> is  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a>
 
<b>by </b><i><a class="txt" href="glib_004.html#E9:56_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E13:56_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V1">[Weighted]</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V2">[ELabeled]</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V3">[VLabeled]</a> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="glib_003.html#NM35">WEVGraphSeq</a> <b>by </b><i><a class="ref" href="glib_003.html#D24">GLIB_003:def 24</a>, <a class="ref" href="glib_003.html#D25">GLIB_003:def 25</a>, <a class="ref" href="glib_003.html#D26">GLIB_003:def 26</a></i>;<br/>
<a NAME="E15:56_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V11">real-WEV</a>
 
<b>by </b><i><a class="txt" href="glib_004.html#E12:56_1_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="glib_003.html#V19">real-WEV</a> <a href="glib_003.html#NM35">WEVGraphSeq</a> <b>by </b><i><a class="ref" href="glib_003.html#D31">GLIB_003:def 31</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E17:56_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#K42">.-&gt;</a> 0 <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K11" target="_self">PRIM:Init</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E5:56_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E18:56_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a>
 
<b>by </b><i><a class="txt" href="glib_004.html#E9:56_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a><b> such that </b><br/><a NAME="E20:56_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E5:56_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>thus </b><a NAME="E21:56_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#K42">.-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E20:56_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>


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