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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">Gk</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>  st <font color="Olive">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">Gk</font>];<br/>
<a NAME="E1:35_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E2:35_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">zero</a>  <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]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:35_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Gk</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>assume </b><a NAME="E1:35_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:35_1_1_1"/>
<font color="Maroon">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:35_1_1_1"/><b>then </b>
( <font color="Maroon">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:35_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">Gk</font>]
 <b>by </b><i><a class="txt" href="chord.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="chord.html#E1:35_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:35_1_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="E3:35_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">k</font> being non <a href="xboole_0.html#V1">zero</a>  <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">k</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(<a class="ref" href="chord.html#T2" target="_self">Th2</a>, );<br/></i>
<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>

<a NAME="E4:35_1"/>
for <font color="Olive">Gk</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>  st <font color="Olive">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">Gk</font>]
 
<b>by </b><i><a class="txt" href="chord.html#E1:35_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:35_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">G</font>]
 <b>by </b><i><a class="txt" href="chord.html#E1:35"><i><font color="Green">E30</font></i></a></i>;<br/>


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