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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:11"/>
( <font color="Maroon">S</font> is <a href="prob_1.html#V3">non-decreasing</a> implies 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</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S</font> is <a href="prob_1.html#V3">non-decreasing</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_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/><a NAME="E1:11_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E2:11_1_1"/>
<font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</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>, <a class="ref" href="prob_1.html#D7">PROB_1:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_1"/>
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</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E38</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">S</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><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>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:11_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>now </b></a><div class="add"><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">S</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E1:11_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><a NAME="E2:11_2_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:11_2_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<a NAME="E2:11_2_1_1"/>
<font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:11_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:11_2_1"/>
for <font color="Olive">k</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">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/></div><b>end;</b></div><b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E5:11_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:11_2"/>
<font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
<font color="Maroon">S</font> is <a href="prob_1.html#V3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="prob_1.html#D7">PROB_1:def 7</a></i>;<br/>


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