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<div class="add">

<a NAME="E1:12_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T32">RCOMP_1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:12_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> <a href="partfun2.html#K4">--&gt;</a> <font color="Maroon">a</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> 1, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E5:12_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E7:12_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E9:12_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:12_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 holds <br/><font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E11:12_1_1"/>
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="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 holds <br/><font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:12_1_1"/><b>then </b>
<font color="Maroon">p</font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T14">FINSEQ_2:14</a></i>;<br/>
<a NAME="E13:12_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E14:12_1_1"/><i><font color="Green">E58</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:12_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E16:12_1_1"/>
<font color="Maroon">p</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_1"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:12_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E3:12_1_1_1"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>proof </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="E1:12_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 ;<br/>

<a NAME="E2:12_1_1_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_1_1"/>
(  not <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> or <font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_1_1_1"/>
<font color="Maroon">p</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:12_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/>


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