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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H</font>];<br/>
<a NAME="E1:111_1"/><i><font color="Green" title="E84">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:111_1_1"/><i><font color="Green" title="E85">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/>for <font color="Olive" title="b2">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">H</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E2:111_1_1"/><i><font color="Green" title="E86">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/><b>assume </b><a NAME="E1:111_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">H</font>
 ;<br/><a NAME="E2:111_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T83" target="_self" title="ZF_LANG:th.83">Th83</a></i>;<br/><b>hence </b><a NAME="E3:111_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">F</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:111_1_1"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="zf_lang.html#E2:111_1_1"><i><font color="Green" title="E86">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:111_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:111"><i><font color="Green" title="E83">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:111_1"/><i><font color="Green" title="E85">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="zf_lang.html#E1:111_1"><i><font color="Green" title="E84">A2</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/>

<a NAME="E3:111_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 
;<br/>
<b>hence </b><a NAME="E4:111_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:111_1"><i><font color="Green" title="E85">A5</font></i></a></i>;<br/>


</div>
