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<a NAME="E1:34_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E2:34_1_1"/><b>then </b><i><font color="Green" title="E12">A1</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E3:34_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">n1</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="triang_1.html#E2:34_1_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E5:34_1_1"/>
 <a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Maroon" title="c3">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K427</a>(<span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span>,<font color="Olive">b<sub>1</sub></font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K427</a>(<span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span>,<font color="Maroon" title="c3">l</font>) )</span><br/><b>let </b><font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span> or  not <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span> or <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K427</a>(<span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span>,<font color="Maroon" title="c4">m</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K427</a>(<span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span>,<font color="Maroon" title="c3">l</font>) )</span><br/>



<b>assume </b><a NAME="E1:34_1_1_1"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c3">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span> &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span> &amp; <font color="Maroon" title="c3">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <a href="seq_1.html#K1" title="SEQ_1:func.1">K427</a>(<span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span>,<font color="Maroon" title="c4">m</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K427</a>(<span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span>,<font color="Maroon" title="c3">l</font>)</span><br/>

<a NAME="E2:34_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E3:34_1_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1_1_1"/>
not <a href="seq_1.html#K1" title="SEQ_1:func.1">K427</a>(<span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span>,<font color="Maroon" title="c4">m</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K427</a>(<span class="p1">(<span class="default"><a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Maroon" title="c2">n1</font></span>)</span>,<font color="Maroon" title="c3">l</font>)
 <b>by </b><i><a class="txt" href="triang_1.html#E1:34_1_1_1"><i><font color="Green" title="E14">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:34_1_1"/><b>then </b>
<font color="Maroon" title="c2">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> :  <a href="triang_1.html#K5" title="TRIANG_1:func.5">@</a> <font color="Olive" title="b1">f</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E7:34_1_1"/>
not <a href="triang_1.html#K6" title="TRIANG_1:func.6">NatEmbSeq</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="triang_1.html#D7" target="_self" title="TRIANG_1:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
