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



<b>assume </b><a NAME="E1:80"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">H</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">F</font>
 ;<br/>

<b>thus </b><a NAME="E2:80"/>
<font color="Maroon">H</font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">F</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="80_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">n</font> = 2; <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">L</font> = <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">F</font></span><a href="finseq_1.html#K10">*&gt;</a></span>;<br/>

<b>thus </b><a NAME="E1:80_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>thus </b><a NAME="E2:80_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>

<b>thus </b><a NAME="E3:80_1"/>
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">H</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">F</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>

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

<b>assume </b><a NAME="E4:80_1"/><i><font color="Green">E32</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E5:80_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E6:80_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:80_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="zf_lang.html#D22" target="_self">Def22</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E8:80_1"/>
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D23" target="_self">Def23</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>

<b>thus </b><a NAME="E9:80_1"/>
<font color="Maroon">H</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#D21" target="_self">Def21</a></i>;<br/>


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

<b>assume </b><a NAME="E3:80"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E4:80"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#D21" target="_self">Def21</a>, <a class="ref" href="zf_lang.html#D15" target="_self">Def15</a></i>;<br/>
<b>hence </b><a NAME="E5:80"/>
contradiction
 ;<br/>


</div>
