<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>





<b>assume </b><a NAME="E1:83"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">F</font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font> &amp; <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D41" target="_self">ZF_LANG:def 41</a><br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">L</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:83"/><i><font color="Green">E32</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 <b>and </b><br/><a NAME="E4:83"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">H1</font>, <font color="Olive">F1</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <br/>( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">F1</font> &amp; <font color="Olive">H1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Olive">F1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E5:83"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T25" target="_self">Th25</a>, <a class="ref" href="zf_lang.html#T26" target="_self">Th26</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E6:83"/><b>then </b><i><font color="Green">E36</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">L'</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:83"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">L'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">L'</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">L'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> )
 <b>and </b><br/><a NAME="E9:83"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/> ex <font color="Olive">H1</font>, <font color="Olive">F1</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <br/>( <font color="Maroon">L'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">L'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">F1</font> &amp; <font color="Olive">H1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Olive">F1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T25" target="_self">Th25</a>, <a class="ref" href="zf_lang.html#D12" target="_self">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:83"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T28" target="_self">Th28</a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <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>reconsider </b><font color="Maroon">L1</font> = <font color="Maroon">L</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/>
<b>thus </b><a NAME="E14:83"/>
<font color="Maroon">F</font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D41" target="_self">ZF_LANG:def 41</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">l</font> = <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D40" target="_self">ZF_LANG:def 40</a><br/>

<b>take </b><font color="Maroon">K</font> = <font color="Maroon">L1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">L'</font>;<br/>

<a NAME="E1:83_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E2:83_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<a NAME="E3:83_1"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E4:83_1"/><b>then </b><i><font color="Green">E43</font></i>: 
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T32" target="_self">Th32</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:83_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T26" target="_self">Th26</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<b>hence </b><a NAME="E6:83_1"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>

<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:83_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E2:83_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:83_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L1</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T26" target="_self">Th26</a>, <a class="ref" href="zf_lang.html#T33" target="_self">Th33</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E4:83_1_1"/><b>then </b>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:83_1_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T37" target="_self">Th37</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:83_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</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="E9:83_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T26" target="_self">Th26</a>, <a class="ref" href="zf_lang.html#T36" target="_self">Th36</a></i>;<br/>

<a NAME="E10:83_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">L1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">L1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T29" target="_self">Th29</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:83_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">L1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T34" target="_self">Th34</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:83_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">L1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T35" target="_self">Th35</a>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>;<br/>
<a NAME="E13:83_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
;<br/>
<b>hence </b><a NAME="E14:83_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T26" target="_self">Th26</a>, <a class="ref" href="zf_lang.html#T29" target="_self">Th29</a>, <a class="ref" href="zf_lang.html#T33" target="_self">Th33</a>, <a class="ref" href="zf_lang.html#T37" target="_self">Th37</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E15:83_1"/><i><font color="Green">E49</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font> )
 ;<br/>

<a NAME="E16:83_1"/>
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E17:83_1"/><b>then </b><i><font color="Green">E50</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:83_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:83_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T33" target="_self">Th33</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:83_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">F1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">G1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E6:83_1_2"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T27" target="_self">Th27</a>, <a class="ref" href="zf_lang.html#T38" target="_self">Th38</a></i>;<br/><b>take </b><font color="Maroon">F1</font> = <font color="Maroon">F1</font>;<br/><b>take </b><font color="Maroon">G1</font> = <font color="Maroon">G1</font>;<br/><b>thus </b><a NAME="E7:83_1_2"/>
( <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T36" target="_self">Th36</a>, <a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, <a class="ref" href="zf_lang.html#T39" target="_self">Th39</a>, , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:83_1_3"/><b>then </b><i><font color="Green">E92</font></i>: 
( 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:83_1_3"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> &amp; <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E4:83_1_3"/>
( <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="zf_lang.html#T36" target="_self">Th36</a>, <a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">F1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">G1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E6:83_1_3"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T27" target="_self">Th27</a>, <a class="ref" href="zf_lang.html#T32" target="_self">Th32</a>, <a class="ref" href="zf_lang.html#T38" target="_self">Th38</a></i>;<br/><b>take </b><font color="Maroon">F1</font> = <font color="Maroon">F1</font>;<br/><b>take </b><font color="Maroon">G1</font> = <font color="Maroon">G1</font>;<br/><b>thus </b><a NAME="E7:83_1_3"/>
( <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T26" target="_self">Th26</a>, <a class="ref" href="zf_lang.html#T29" target="_self">Th29</a>, <a class="ref" href="zf_lang.html#T32" target="_self">Th32</a>, <a class="ref" href="zf_lang.html#T34" target="_self">Th34</a>, <a class="ref" href="zf_lang.html#T35" target="_self">Th35</a>, <a class="ref" href="zf_lang.html#T36" target="_self">Th36</a>, <a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, , <a class="ref" href="zf_lang.html#T46" target="_self">Th46</a>, , , <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_4"/><i><font color="Green">E95</font></i>: 
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:83_1_4"/><b>then </b><i><font color="Green">E96</font></i>: 
( 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:83_1_4"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">j1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:83_1_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">j1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:83_1_4"/><i><font color="Green">E98</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> &amp; <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> )
 ;<br/><i><font color="Green">E99</font></i>: <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">L1</font></span>)</span> = 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">L1</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="zf_lang.html#T26" target="_self">Th26</a>, <a class="ref" href="zf_lang.html#T33" target="_self">Th33</a>, , , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>
;<br/><a NAME="E9:83_1_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then consider </b><font color="Maroon">F1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">G1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E11:83_1_4"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">L'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">L'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T31" target="_self">Th31</a>, </i>;<br/><b>take </b><font color="Maroon">F1</font> = <font color="Maroon">F1</font>;<br/><b>take </b><font color="Maroon">G1</font> = <font color="Maroon">G1</font>;<br/><b>thus </b><a NAME="E12:83_1_4"/>
( <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T34" target="_self">Th34</a>, <a class="ref" href="zf_lang.html#T35" target="_self">Th35</a>, , <a class="ref" href="zf_lang.html#T46" target="_self">Th46</a>, , <a class="txt" href="zf_lang.html#E19:83_1"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:83_1"/>
 ex <font color="Olive">H1</font>, <font color="Olive">F1</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <br/>( <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">F1</font> &amp; <font color="Olive">H1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Olive">F1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T40" target="_self">Th40</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


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

<b>assume </b><a NAME="E15:83"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E16:83"/>
( <font color="Maroon">F</font> <a href="zf_lang.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T25" target="_self">Th25</a>, <a class="ref" href="zf_lang.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E17:83"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#D21" target="_self">Def21</a></i>;<br/>
<b>hence </b><a NAME="E18:83"/>
contradiction
 <b>by </b><i><a class="ref" href="zf_lang.html#T33" target="_self">Th33</a></i>;<br/>


</div>
