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<div class="add">

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

<b>assume </b><a NAME="E1:32"/>
not <font color="Maroon">H</font> is <a href="zf_lang.html#V7" target="_self">atomic</a>
 ;<br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">H</font> is <a href="zf_lang.html#V4" target="_self">negative</a> or <font color="Maroon">H</font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a> or <font color="Maroon">H</font> is <a href="zf_lang.html#V6" target="_self">universal</a> )
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V4" target="_self">negative</a>
 ;<br/><b>then consider </b><font color="Maroon">H1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">H1</font> = <font color="Maroon">H1</font>;<br/><a NAME="E4:32_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/><a NAME="E5:32_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E6:32_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>assume </b><a NAME="E1:32_2"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a>
 ;<br/><b>then consider </b><font color="Maroon">H1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">F1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:32_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Maroon">F1</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">H1</font> = <font color="Maroon">H1</font>;<br/><a NAME="E4:32_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F1</font>
 <b>by </b><i/>;<br/><a NAME="E5:32_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E6:32_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>assume </b><a NAME="E1:32_3"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V6" target="_self">universal</a>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">H1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:32_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">H1</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">H1</font> = <font color="Maroon">H1</font>;<br/><a NAME="E4:32_3"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/><a NAME="E5:32_3"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">4,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="zf_lang.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">4,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span></span>)</span> &amp; 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</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> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/><b>hence </b><a NAME="E6:32_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
 ex <font color="Olive">H1</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">H1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i>, , </i>;<br/>


</div>
