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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(),  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>(),   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ] <b>means </b>( ( <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a>  st <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">p</font>) ) &amp; ( for <font color="Olive">A</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">A</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span>) ) &amp; ( for <font color="Olive">A</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">A</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span>) ) &amp; ( for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">B</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</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">A</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/>( ( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">p</font>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">B</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">B</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">C</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">F</font>,<font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E1:76_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>();<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E2:76_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E3:76_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>hence </b><a NAME="E4:76_1_1"/>
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">p</font>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">B</font> holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">B</font> holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">C</font> holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font></span>)</span>) ) )
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:76_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>proof </b></a><div class="add">

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

<b>given </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><a NAME="E1:76_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F</font>,<font color="Maroon">k</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()] <b>means </b>( (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">F</font>,<font color="Maroon">a<sub>1</sub></font>] ) );<br/>
<a NAME="E2:76_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_2_1_1_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  ) or (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp;  ex <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> ) or (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp;  ex <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">B</font> ) or (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp;  ex <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">B</font> ) or (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp;  ex <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">C</font> ) )
 </span><b>by </b><i><a class="ref" href="modal_1.html#T10" target="_self">Th10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:76_1_2_1_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:76_1_2_1_1_1_2"/><i><font color="Green">E61</font></i>: 
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  )
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>2</sub></font>();<br/><b>thus </b><a NAME="E2:76_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">F</font>,<font color="Maroon">A</font>]
 <b>by </b><i>, <a class="ref" href="modal_1.html#T15" target="_self">Th15</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:76_1_2_1_1_1_3"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp;  ex <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="modal_1.html#NM1">MP-variable</a><b> such that </b><br/><a NAME="E3:76_1_2_1_1_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">p</font>
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">p</font>);<br/><b>thus </b><a NAME="E4:76_1_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">F</font>,<font color="Maroon">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:76_1_2_1_1_1_3"><i><font color="Green">E62</font></i></a>, , <a class="txt" href="modal_1.html#E1:76_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="modal_1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:76_1_2_1_1_1_4"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp;  ex <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">B</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">B</font> being   <a href="modal_1.html#NM5">MP-wff</a><b> such that </b><br/><a NAME="E3:76_1_2_1_1_1_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">B</font>
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>);<br/><b>thus </b><a NAME="E4:76_1_2_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">F</font>,<font color="Maroon">A</font>]
 <b>by </b><i>, <a class="ref" href="modal_1.html#T15" target="_self">Th15</a>, <a class="txt" href="modal_1.html#E2:76_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="modal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="modal_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1_1_1_5"><b>case </b></a><a NAME="E1:76_1_2_1_1_1_5"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp;  ex <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">B</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">B</font> being   <a href="modal_1.html#NM5">MP-wff</a><b> such that </b><br/><a NAME="E3:76_1_2_1_1_1_5"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">B</font>
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>);<br/><b>thus </b><a NAME="E4:76_1_2_1_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">F</font>,<font color="Maroon">A</font>]
 <b>by </b><i>, <a class="ref" href="modal_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="modal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="modal_1.html#T12" target="_self">Th12</a>, <a class="ref" href="modal_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_1_1_1_6"><b>case </b></a><a NAME="E1:76_1_2_1_1_1_6"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp;  ex <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">B</font> being   <a href="modal_1.html#NM5">MP-wff</a>, <font color="Maroon">C</font> being   <a href="modal_1.html#NM5">MP-wff</a><b> such that </b><br/><a NAME="E3:76_1_2_1_1_1_6"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">C</font>
 ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">C</font></span>)</span>);<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_1_1_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_1_1_1_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B1</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/><b>let </b><font color="Maroon">C1</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/><b>assume </b><a NAME="E1:76_1_2_1_1_1_6_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">C1</font>
 ;<br/><a NAME="E2:76_1_2_1_1_1_6_1_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> )
 <b>by </b><i>, <a class="txt" href="modal_1.html#E1:76_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><b>hence </b><a NAME="E3:76_1_2_1_1_1_6_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">C1</font></span>)</span>)
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:76_1_2_1_1_1_6_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">F</font>,<font color="Maroon">A</font>]
 <b>by </b><i>, <a class="ref" href="modal_1.html#T15" target="_self">Th15</a>, <a class="ref" href="modal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="modal_1.html#T12" target="_self">Th12</a>, <a class="ref" href="modal_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:76_1_2_1_1_1_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">F</font>,<font color="Maroon">A</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:76_1_2_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> ) &amp; (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Maroon">F</font>,<font color="Maroon">A</font>] ) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:76_1_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">p</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">p</font>,<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>take </b><font color="Maroon">H</font> = <font color="Maroon">G</font>;<br/>

<b>thus </b><a NAME="E5:76_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">H</font>,<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ;<br/>

<b>set </b><font color="Maroon">p</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:76_1_2_2"/><i><font color="Green">E71</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>thus </b><a NAME="E2:76_1_2_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_2_2_1_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_2_2_1_1_1"/><i><font color="Green">E76</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:76_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:76_1_2_2_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:76_1_2_2_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i>, <a class="ref" href="modal_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_2_2_1_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_1_2_2_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  implies <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E3:76_1_2_2"/>
for <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">p</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="modal_1.html#NM1">MP-variable</a>;<br/>

<b>assume </b><a NAME="E1:76_1_2_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">q</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_2_2_2_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:76_1_2_2_2_1_1"/><i><font color="Green">E79</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:76_1_2_2_2_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:76_1_2_2_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="modal_1.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E4:76_1_2_2_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">q</font>)
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:76_1_2_2_2_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_1_2_2_2_1_2"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">q</font>)
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E4:76_1_2_2"/>
for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">B</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E1:76_1_2_2_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:76_1_2_2_3"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="txt" href="modal_1.html#E2:76_1_2"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E3:76_1_2_2_3"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_2_2_3_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_3_1_1"><b>suppose </b></a><a NAME="E1:76_1_2_2_3_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:76_1_2_2_3_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:76_1_2_2_3_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:76_1_2_2_3_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>)
 <b>by </b><i>, , <a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_3_1_2"><b>suppose </b></a><a NAME="E1:76_1_2_2_3_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_1_2_2_3_1_2"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>)
 <b>by </b><i>, , <a class="ref" href="modal_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E5:76_1_2_2"/>
for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">B</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E1:76_1_2_2_4"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:76_1_2_2_4"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:76_1_2_2_4"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_2_2_4_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_4_1_1"><b>suppose </b></a><a NAME="E1:76_1_2_2_4_1_1"/><i><font color="Green">E132</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:76_1_2_2_4_1_1"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:76_1_2_2_4_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:76_1_2_2_4_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>)
 <b>by </b><i>, , <a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_4_1_2"><b>suppose </b></a><a NAME="E1:76_1_2_2_4_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_1_2_2_4_1_2"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>)
 <b>by </b><i>, , <a class="ref" href="modal_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E6:76_1_2_2"/>
for <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">C</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/>



<b>assume </b><a NAME="E1:76_1_2_2_5"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:76_1_2_2_5"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:76_1_2_2_5"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:76_1_2_2_5"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:76_1_2_2_5"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_2_2_5_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_5_1_1"><b>suppose </b></a><a NAME="E1:76_1_2_2_5_1_1"/><i><font color="Green">E138</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:76_1_2_2_5_1_1"/><b>then </b><i><font color="Green">E139</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:76_1_2_2_5_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:76_1_2_2_5_1_1"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">C</font></span>)</span>)
 <b>by </b><i>, , <a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2_2_5_1_2"><b>suppose </b></a><a NAME="E1:76_1_2_2_5_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_1_2_2_5_1_2"/>
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">C</font></span>)</span>)
 <b>by </b><i>, , <a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


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


</div><b>end;</b></div>
<a NAME="E3:76_1"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="modal_1.html#T16" target="_self">Th16</a>, );<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">A</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">A</font></span>)</span>] holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font> ) );<br/>
<a NAME="E4:76_1"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:76_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="modal_1.html#K6" target="_self">MP-WFF</a> 
 <b>and </b><br/><a NAME="E2:76_1_3"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>]
 <b>and </b><br/><a NAME="E3:76_1_3"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>]
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> <b> such that </b><br/><a NAME="E5:76_1_3"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:76_1_3"/><i><font color="Green">E187</font></i>: 
for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span>] holds <br/><font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E8:76_1_3"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span>]
 <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">y1</font> = 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:76_1"/><i><font color="Green">E189</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="modal_1.html#K6" target="_self">MP-WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  ;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:76_1_4"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font></span>)</span>]
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
;<br/>

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

<b>thus </b><a NAME="E5:76_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 ;<br/>

<b>let </b><font color="Maroon">G</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E6:76_1_4"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font></span>)</span>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ] <b>means </b>(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font></span>)</span> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E7:76_1_4"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K17" target="_self">VERUM</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1_4_1"/><i><font color="Green">E193</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="modal_1.html#K17" target="_self">VERUM</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font></span>)</span>
 ;<br/>

<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <a href="modal_1.html#K17" target="_self">VERUM</a>  = 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <a href="modal_1.html#K17" target="_self">VERUM</a> 
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E8:76_1_4"/><i><font color="Green">E194</font></i>: 
for <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a> holds  <font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="modal_1.html#NM1">MP-variable</a>;<br/>

<b>assume </b><a NAME="E1:76_1_4_2"/><i><font color="Green">E195</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font></span>)</span>
 ;<br/>

<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">p</font></span>)</span> = 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">p</font>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E9:76_1_4"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">A</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E1:76_1_4_3"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">A</font>]
 ;<br/>

<b>assume </b><a NAME="E2:76_1_4_3"/><i><font color="Green">E198</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font></span>)</span>
 ;<br/>

<a NAME="E3:76_1_4_3"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span> = 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span>)
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E10:76_1_4"/><i><font color="Green">E199</font></i>: 
for <font color="Olive">A</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E1:76_1_4_4"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">A</font>]
 ;<br/>

<b>assume </b><a NAME="E2:76_1_4_4"/><i><font color="Green">E201</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font></span>)</span>
 ;<br/>

<a NAME="E3:76_1_4_4"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span>)
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E11:76_1_4"/><i><font color="Green">E202</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">A</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="modal_1.html#NM5">MP-wff</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:76_1_4_5"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">A</font>]
 <b>and </b><br/><a NAME="E2:76_1_4_5"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">B</font>]
 <b>and </b><br/><a NAME="E3:76_1_4_5"/><i><font color="Green">E205</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font></span>)</span>
 ;<br/>

<a NAME="E4:76_1_4_5"/><i><font color="Green">E206</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:76_1_4_5"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span> = 
<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>)
<b>by </b><i>, , , <a class="ref" href="modal_1.html#T18" target="_self">Th18</a>, <a class="ref" href="modal_1.html#T19" target="_self">Th19</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, <a class="ref" href="modal_1.html#T18" target="_self">Th18</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E12:76_1_4"/>
for <font color="Olive">p</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="modal_1.html#S1" target="_self">MODAL_1:sch 1</a>(<a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, , , , );<br/></i>
<b>hence </b><a NAME="E13:76_1_4"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:76_1"/><i><font color="Green">E207</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K6" target="_self">MP-WFF</a> 
 <b>and </b><br/><a NAME="E8:76_1"/><i><font color="Green">E208</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E9:76_1"/>
<font color="Maroon">F</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:76_1_5"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_1_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:76_1_5_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="modal_1.html#K6" target="_self">MP-WFF</a> 
 <b>and </b><br/><a NAME="E4:76_1_5_1"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> <b> such that </b><br/><a NAME="E6:76_1_5_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E7:76_1_5_1"/><i><font color="Green">E211</font></i>: 
for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span>] holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, </i>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E9:76_1_5_1"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span>]
 <b>by </b><i/>;<br/>
<a NAME="E10:76_1_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:76_1_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:76_1_5"/>
<font color="Maroon">F</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="modal_1.html#NM5">MP-wff</a><b> such that </b><br/><a NAME="E12:76_1"/><i><font color="Green">E213</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <a href="modal_1.html#K17" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E14:76_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>]
 <b>by </b><i/>;<br/>
<a NAME="E15:76_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <a href="modal_1.html#K17" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <a href="modal_1.html#K17" target="_self">VERUM</a> 
 
<b>by </b><i><a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, </i>;<br/>
<b>hence </b><a NAME="E16:76_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <a href="modal_1.html#K17" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="ref" href="modal_1.html#T17" target="_self">Th17</a>, </i>;<br/>

<b>thus </b><a NAME="E17:76_1"/>
for <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a> holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">p</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="modal_1.html#NM1">MP-variable</a>;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="modal_1.html#NM5">MP-wff</a><b> such that </b><br/><a NAME="E2:76_1_6"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:76_1_6"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>]
 <b>by </b><i/>;<br/>
<a NAME="E5:76_1_6"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:76_1_6"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">p</font>)
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E18:76_1"/>
for <font color="Olive">A</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ;<br/>

<b>consider </b><font color="Maroon">A1</font> being   <a href="modal_1.html#NM5">MP-wff</a><b> such that </b><br/><a NAME="E2:76_1_7"/><i><font color="Green">E217</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A1</font></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:76_1_7"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A1</font></span>)</span>]
 <b>by </b><i/>;<br/>
<a NAME="E5:76_1_7"/><i><font color="Green">E219</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>,<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_7_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:76_1_7_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:76_1_7_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:76_1_7_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:76_1_7_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  implies <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">p</font>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">B</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">B</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">C</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font></span>)</span>) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:76_1_7"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">k</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>;<br/>
<a NAME="E7:76_1_7"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:76_1_7"/><b>then </b><i><font color="Green">E222</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E9:76_1_7"/>
 ex <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">B</font></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E10:76_1_7"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:76_1_7"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" target="_self">'not'</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span>)
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E19:76_1"/>
for <font color="Olive">A</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ;<br/>

<b>consider </b><font color="Maroon">A1</font> being   <a href="modal_1.html#NM5">MP-wff</a><b> such that </b><br/><a NAME="E2:76_1_8"/><i><font color="Green">E223</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A1</font></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:76_1_8"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A1</font></span>)</span>]
 <b>by </b><i/>;<br/>
<a NAME="E5:76_1_8"/><i><font color="Green">E225</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>,<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_8_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:76_1_8_1"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:76_1_8_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:76_1_8_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:76_1_8_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  implies <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">p</font>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">B</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">B</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">C</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font></span>)</span>) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:76_1_8"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">k</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>;<br/>
<a NAME="E7:76_1_8"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:76_1_8"/><b>then </b><i><font color="Green">E228</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E9:76_1_8"/>
 ex <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">B</font></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E10:76_1_8"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:76_1_8"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" target="_self">(#)</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span>)
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E20:76_1"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> , <font color="Maroon">B</font> be    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ;<br/>

<b>consider </b><font color="Maroon">A1</font> being   <a href="modal_1.html#NM5">MP-wff</a><b> such that </b><br/><a NAME="E2:76_1_9"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A1</font></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:76_1_9"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A1</font></span>)</span>]
 <b>by </b><i/>;<br/>
<a NAME="E5:76_1_9"/><i><font color="Green">E231</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>,<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_9_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:76_1_9_1"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="modal_1.html#M2" target="_self">Element</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:76_1_9_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:76_1_9_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:76_1_9_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K17" target="_self">VERUM</a>  implies <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive">p</font> being  <a href="modal_1.html#NM1">MP-variable</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K16" target="_self">@</a> <font color="Olive">p</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">p</font>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K10" target="_self">'not'</a> <font color="Olive">B</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="modal_1.html#K11" target="_self">(#)</a> <font color="Olive">B</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>) ) &amp; ( for <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="modal_1.html#NM5">MP-wff</a>  st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Olive">C</font> holds <br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font></span>)</span>) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:76_1_9"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">k1</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">k2</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font></span>)</span>;<br/>
<a NAME="E7:76_1_9"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:76_1_9"/><b>then </b><i><font color="Green">E236</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font></span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E9:76_1_9"/>
 ex <font color="Olive">B1</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <br/>( <font color="Olive">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">B1</font></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B1</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E10:76_1_9"/><b>then </b><i><font color="Green">E237</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:76_1_9"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:76_1_9"/><b>then </b><i><font color="Green">E238</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">G</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font></span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E13:76_1_9"/>
 ex <font color="Olive">C</font> being  <a href="modal_1.html#NM5">MP-wff</a> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modal_1.html#K6" target="_self">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>, <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">C</font></span>)</span>] holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">C</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E14:76_1_9"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="modal_1.html#T19" target="_self">Th19</a></i>;<br/>
<b>hence </b><a NAME="E15:76_1_9"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="modal_1.html#K12" target="_self">'&amp;'</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font></span>)</span>)
 <b>by </b><i>, , , <a class="ref" href="modal_1.html#T18" target="_self">Th18</a></i>;<br/>


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


</div>
