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

<b>let </b><font color="Maroon">M</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">H1</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">v1</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font>  st not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">M</font>,<font color="Olive">v1</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/> ex <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> ex <font color="Olive">v2</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font> st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>2</sub></font> &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Olive">H2</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4 ) &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H2</font> &amp; <font color="Maroon">M</font>,<font color="Olive">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Olive">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Olive">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Olive">H2</font>,<font color="Olive">v2</font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>,<font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E1:17"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </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>let </b><font color="Maroon">v1</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font>;<br/>



<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E47</font></i>: 
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; <font color="Maroon">M</font>,<font color="Maroon">v1</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 ;<br/>

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

<b>take </b>
<font color="Maroon">v1</font>
;<br/>

<b>thus </b><a NAME="E2:17_1"/>
( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4 ) &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; <font color="Maroon">M</font>,<font color="Maroon">v1</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:17"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>,<font color="Maroon">i</font>]
 ;<br/>

<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">v1</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font>;<br/>



<b>assume </b><a NAME="E2:17_2"/><i><font color="Green">E50</font></i>: 
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; <font color="Maroon">M</font>,<font color="Maroon">v1</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E3:17_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 implies  ex <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> ex <font color="Olive">v2</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font> st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Olive">H2</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4 ) &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H2</font> &amp; <font color="Maroon">M</font>,<font color="Olive">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Olive">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Olive">H2</font>,<font color="Olive">v2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 )
 ;<br/>

<b>consider </b><font color="Maroon">H1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">v'</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:17_2_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4
 <b>and </b><br/><a NAME="E4:17_2_1"/><i><font color="Green">E54</font></i>: 
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">M</font>,<font color="Maroon">v'</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E5:17_2_1"/><i><font color="Green">E55</font></i>: 
 <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H1</font>,<font color="Maroon">v'</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:17_2_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K14">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">H1</font></span>)</span> holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:17_2_1"/>
( not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K14">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">H1</font></span>)</span> &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K14">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">H1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="zf_lang1.html#T160">ZF_LANG1:160</a></i>;<br/>
<a NAME="E9:17_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font> &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K14">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">H1</font></span>)</span> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K14">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">H1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:17_2_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 
<b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">H2</font> = <font color="Maroon">H1</font> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span>;<br/>

<b>take </b><font color="Maroon">v2</font> = <font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span>)</span>;<br/>

<a NAME="E11:17_2_1"/>
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> &amp; 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> &amp; 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:17_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> &amp;  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> &amp;  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> &amp;  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> &amp;  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/>
<a NAME="E13:17_2_1"/><i><font color="Green">E61</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 &amp;  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 &amp;  <a href="zf_lang.html#K2">x.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/>
<a NAME="E14:17_2_1"/>
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E15:17_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:17_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font>
 
;<br/>
<b>thus </b><a NAME="E17:17_2_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H2</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1"><b>proof </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="E1:17_2_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:17_2_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:17_2_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E4:17_2_1_1"/>
 <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:17_2_1_1"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:17_2_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_lang1.html#T201">ZF_LANG1:201</a></i>;<br/>
<b>assume </b><a NAME="E7:17_2_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H2</font>
 ;<br/>

<a NAME="E8:17_2_1_1"/><b>then </b>
(  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E9:17_2_1_1"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; (  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> or  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span> ) &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/>
<a NAME="E10:17_2_1_1"/><b>then </b>
 <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:17_2_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4 )
 <b>by </b><i>, , , </i>;<br/>


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

<b>thus </b><a NAME="E18:17_2_1"/>
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/>

<i><font color="Green">E69</font></i>: <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span> = 
 <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="zf_lang1.html#T173">ZF_LANG1:173</a></i>
<br/>.= 
 <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="zf_lang1.html#T178">ZF_LANG1:178</a></i>
<br/>.= 
 <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="zf_lang1.html#T173">ZF_LANG1:173</a></i>
<br/>.= 
 <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T177">ZF_LANG1:177</a></i>
<br/>.= 
 <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="zf_lang1.html#T166">ZF_LANG1:166</a></i>
;<br/>
<i><font color="Green">E70</font></i>:  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T155">ZF_LANG1:155</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T159">ZF_LANG1:159</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T155">ZF_LANG1:155</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T158">ZF_LANG1:158</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T151">ZF_LANG1:151</a></i>
;<br/>
<a NAME="E21:17_2_1"/>
not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E22:17_2_1"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E23:17_2_1"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E24:17_2_1"/><b>then </b>
( not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E25:17_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
( not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp; not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p0">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_lang1.html#T198">ZF_LANG1:198</a></i>;<br/>
<a NAME="E26:17_2_1"/><b>then </b>
( <font color="Maroon">M</font>,<font color="Maroon">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">v'</font> &amp; <font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="zf_lang1.html#T78">ZF_LANG1:78</a>, <a class="ref" href="zf_lang1.html#T79">ZF_LANG1:79</a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E27:17_2_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">M</font>,<font color="Maroon">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_2_1_4"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> ;<br/><a NAME="E3:17_2_1_4"/>
<font color="Maroon">M</font>,<font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="zf_lang1.html#T80">ZF_LANG1:80</a></i>;<br/><b>then consider </b><font color="Maroon">m'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E5:17_2_1_4"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T82">ZF_LANG1:82</a></i>;<br/><a NAME="E6:17_2_1_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> )
 <b>by </b><i>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/><a NAME="E7:17_2_1_4"/><b>then </b><i><font color="Green">E74</font></i>: 
( <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> &amp; <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span> )
 <b>by </b><i>, <a class="ref" href="zf_lang1.html#T80">ZF_LANG1:80</a>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/><a NAME="E8:17_2_1_4"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T19">ZF_MODEL:19</a></i>;<br/><a NAME="E9:17_2_1_4"/><i><font color="Green">E76</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="zf_lang1.html#T79">ZF_LANG1:79</a></i>;<br/><a NAME="E10:17_2_1_4"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H1</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E11:17_2_1_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</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"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</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"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font> &amp; <font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">v'</font> )
 <b>by </b><i>, , , <a class="ref" href="zf_lang1.html#T78">ZF_LANG1:78</a>, <a class="ref" href="zf_lang1.html#T79">ZF_LANG1:79</a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/><a NAME="E12:17_2_1_4"/><b>then </b>
( <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">m'</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H2</font> )
 <b>by </b><i>, , , , , <a class="ref" href="zf_model.html#T19">ZF_MODEL:19</a></i>;<br/><a NAME="E13:17_2_1_4"/><b>then </b>
( <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">m'</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H2</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E14:17_2_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H2</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> &amp; <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H1</font>,<font color="Maroon">v'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E15:17_2_1_4"/>
<span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H2</font>,<font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H1</font>,<font color="Maroon">v'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:17_2_1"/>
 <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H2</font>,<font color="Maroon">v2</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_2"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 4 )
 ;<br/><b>thus </b><a NAME="E2:17_2_2"/>
 ex <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> ex <font color="Olive">v2</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font> st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Olive">H2</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4 ) &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H2</font> &amp; <font color="Maroon">M</font>,<font color="Olive">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Olive">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Olive">H2</font>,<font color="Olive">v2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">H2</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">v2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font><b> such that </b><br/><a NAME="E2:17_2_2_1"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H2</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4
 <b>and </b><br/><a NAME="E3:17_2_2_1"/><i><font color="Green">E104</font></i>: 
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">M</font>,<font color="Maroon">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E4:17_2_2_1"/><i><font color="Green">E105</font></i>: 
 <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H2</font>,<font color="Maroon">v2</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">H2</font>
;<br/>

<b>take </b>
<font color="Maroon">v2</font>
;<br/>

<b>thus </b><a NAME="E5:17_2_2_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H2</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_1_1"><b>proof </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="E1:17_2_2_1_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H2</font> )
 ;<br/>

<a NAME="E2:17_2_2_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:17_2_2_1_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:17_2_2_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4 )
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E6:17_2_2_1"/>
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">M</font>,<font color="Maroon">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H2</font>,<font color="Maroon">v2</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:17_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 implies  ex <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> ex <font color="Olive">v2</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font> st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Olive">H2</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4 ) &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H2</font> &amp; <font color="Maroon">M</font>,<font color="Olive">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Olive">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Olive">H2</font>,<font color="Olive">v2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2_3"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:17_2_3"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font> holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:17_2_3"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4 or  not <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4 )
 
;<br/>
<b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:17_2_3"/><i><font color="Green">E110</font></i>: 
( ( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4 &amp; <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) or ( not <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4 &amp; <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 5 ) )
 ;<br/>
<b>take </b><font color="Maroon">F</font> = <font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k1</font></span>)</span>;<br/>

<b>take </b>
<font color="Maroon">v1</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span>
;<br/>

<a NAME="E7:17_2_3"/>
( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 &amp; <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 )
 
<b>by </b><i/>;<br/>
<a NAME="E8:17_2_3"/><b>then </b><i><font color="Green">E111</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 0 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/>
<a NAME="E9:17_2_3"/><i><font color="Green">E112</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2_3_1"/>
 <a href="zf_lang.html#K2">x.</a> <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:17_2_3_1"/><b>then </b>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:17_2_3_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E10:17_2_3"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">F</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_3_2"><b>proof </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="E1:17_2_3_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:17_2_3_2"/><b>then </b>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:17_2_3_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>hence </b><a NAME="E4:17_2_3_2"/>
(  not  <a href="zf_lang.html#K2">x.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">F</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4 )
 <b>by </b><i>, <a class="ref" href="zf_lang1.html#T198">ZF_LANG1:198</a></i>;<br/>


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

<a NAME="E11:17_2_3"/>
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 &amp;  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/>
<b>hence </b><a NAME="E12:17_2_3"/>
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">F</font> &amp; <font color="Maroon">M</font>,<font color="Maroon">v1</font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> <font color="Maroon">k1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:17_2"/>
 ex <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> ex <font color="Olive">v2</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font> st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Olive">H2</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4 ) &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H2</font> &amp; <font color="Maroon">M</font>,<font color="Olive">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Olive">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Olive">H2</font>,<font color="Olive">v2</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:17"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:17"/>
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">M</font>,<font color="Maroon">v1</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> implies  ex <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> ex <font color="Olive">v2</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">M</font> st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp;  <a href="zf_lang.html#K2">x.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Olive">H2</font> &amp; not <font color="Olive">j</font> <a href="hidden.html#R1">=</a> 3 holds <br/><font color="Olive">j</font> <a href="hidden.html#R1">=</a> 4 ) &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H2</font> &amp; <font color="Maroon">M</font>,<font color="Olive">v2</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Olive">H2</font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">H1</font>,<font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Olive">H2</font>,<font color="Olive">v2</font> ) )
 ;<br/>


</div>
