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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(),  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>(),   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ] <b>means </b>( ( $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  implies $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>  st $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> holds <br/>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">A</font> holds <br/>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">A</font> holds <br/>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">B</font> holds <br/>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span>,<span class="p1">(<span class="default">$2 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">B</font></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>(),   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">A</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $2 holds <br/>( ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  implies $1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive" title="b2">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b2">p</font> holds <br/>$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">p</font>) ) &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b2">B</font> holds <br/>$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b2">B</font> holds <br/>$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b3">C</font> holds <br/>$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">B</font></span>)</span>,<span class="p1">(<span class="default">$1 <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">C</font></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">F</font>,$1];<br/>
<a NAME="E1:69_1"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c1">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>();<br/>
<b>take </b>
<font color="Maroon" title="c1">F</font>
;<br/>

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E2:69_1_1"/><i><font color="Green" title="E52">A2</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E3:69_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:69_1_1"/>
( ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  implies <font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">C</font></span>)</span>) ) )
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:69_1"/><i><font color="Green" title="E52">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>given </b><font color="Maroon" title="c2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><a NAME="E2:69_1_2"/><i><font color="Green" title="E53">A4</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">k</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()] <b>means </b>( (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> $1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 ) &amp; (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[$2,<font color="Maroon" title="c2">F</font>,$1] ) );<br/>
<a NAME="E3:69_1_2"/><i><font color="Green" title="E54">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>   ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font> be    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_2_1_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  ) or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> ) or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> ) or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> ) or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> ) )
 </span><b>by </b><i><a class="ref" href="modal_1.html#T48" target="_self" title="MODAL_1:th.48">Th48</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:69_1_2_1_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">A</font>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:69_1_2_1_1_1_2"/><i><font color="Green" title="E55">A6</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>2</sub></font>();<br/><b>thus </b><a NAME="E2:69_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_1_1_1_2"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="modal_1.html#E75"><i><font color="Green" title="E48">Lm4</font></i></a>, <a class="txt" href="modal_1.html#E77"><i><font color="Green" title="E50">Lm6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:69_1_2_1_1_1_3"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">p</font> being   <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a><b> such that </b><br/><a NAME="E3:69_1_2_1_1_1_3"/><i><font color="Green" title="E55">A7</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c4">p</font>
 ;<br/><b>take </b><font color="Maroon" title="c5">y</font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">p</font>);<br/><b>thus </b><a NAME="E4:69_1_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1_2_1_1_1_3"><i><font color="Green" title="E55">A7</font></i></a>, <a class="txt" href="modal_1.html#E77"><i><font color="Green" title="E50">Lm6</font></i></a>, <a class="ref" href="modal_1.html#T37" target="_self" title="MODAL_1:th.37">Th37</a>, <a class="ref" href="modal_1.html#T49" target="_self" title="MODAL_1:th.49">Th49</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:69_1_2_1_1_1_4"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">B</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><b> such that </b><br/><a NAME="E3:69_1_2_1_1_1_4"/><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c4">B</font>
 ;<br/><b>take </b><font color="Maroon" title="c5">y</font> = <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">B</font></span>)</span>);<br/><b>thus </b><a NAME="E4:69_1_2_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1_2_1_1_1_4"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="modal_1.html#E75"><i><font color="Green" title="E48">Lm4</font></i></a>, <a class="ref" href="modal_1.html#T38" target="_self" title="MODAL_1:th.38">Th38</a>, <a class="ref" href="modal_1.html#T49" target="_self" title="MODAL_1:th.49">Th49</a>, <a class="ref" href="modal_1.html#T50" target="_self" title="MODAL_1:th.50">Th50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1_5"><b>case </b></a><a NAME="E1:69_1_2_1_1_1_5"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">B</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><b> such that </b><br/><a NAME="E3:69_1_2_1_1_1_5"/><i><font color="Green" title="E55">A9</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c4">B</font>
 ;<br/><b>take </b><font color="Maroon" title="c5">y</font> = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">B</font></span>)</span>);<br/><b>thus </b><a NAME="E4:69_1_2_1_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1_2_1_1_1_5"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="modal_1.html#E75"><i><font color="Green" title="E48">Lm4</font></i></a>, <a class="ref" href="modal_1.html#T39" target="_self" title="MODAL_1:th.39">Th39</a>, <a class="ref" href="modal_1.html#T49" target="_self" title="MODAL_1:th.49">Th49</a>, <a class="ref" href="modal_1.html#T50" target="_self" title="MODAL_1:th.50">Th50</a>, <a class="ref" href="modal_1.html#T51" target="_self" title="MODAL_1:th.51">Th51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1_6"><b>case </b></a><a NAME="E1:69_1_2_1_1_1_6"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><b> such that </b><br/><a NAME="E3:69_1_2_1_1_1_6"/><i><font color="Green" title="E55">A10</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c5">C</font>
 ;<br/><b>take </b><font color="Maroon" title="c6">y</font> = <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">C</font></span>)</span>);<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_1_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">B1</font>, <font color="Maroon" title="c8">C1</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/><b>assume </b><a NAME="E1:69_1_2_1_1_1_6_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B1</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c8">C1</font>
 ;<br/><a NAME="E2:69_1_2_1_1_1_6_1_1"/><b>then </b>
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B1</font> &amp; <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C1</font> )
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1_2_1_1_1_6"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="modal_1.html#T40" target="_self" title="MODAL_1:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E3:69_1_2_1_1_1_6_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">C1</font></span>)</span>)
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:69_1_2_1_1_1_6_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1_2_1_1_1_6"><i><font color="Green" title="E55">A10</font></i></a>, <a class="txt" href="modal_1.html#E75"><i><font color="Green" title="E48">Lm4</font></i></a>, <a class="ref" href="modal_1.html#T49" target="_self" title="MODAL_1:th.49">Th49</a>, <a class="ref" href="modal_1.html#T50" target="_self" title="MODAL_1:th.50">Th50</a>, <a class="ref" href="modal_1.html#T51" target="_self" title="MODAL_1:th.51">Th51</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:69_1_2_1_1_1_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:69_1_2_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">A</font> ) &amp; (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>] ) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:69_1_2"/><i><font color="Green" title="E55">A11</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c3">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="modal_1.html#E3:69_1_2"><i><font color="Green" title="E54">A5</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c4">H</font> = <font color="Maroon" title="c3">G</font>;<br/>

<b>thus </b><a NAME="E6:69_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">H</font>,<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ;<br/>

<b>set </b><font color="Maroon" title="c6">p</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:69_1_2_2"/><i><font color="Green" title="E56">A12</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<b>thus </b><a NAME="E2:69_1_2_2"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_2_2_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_2_1_1_1"/><i><font color="Green" title="E57">A13</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E58">A14</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:69_1_2_2_1_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_1_1_1"><i><font color="Green" title="E57">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_1_2_2_1_1_1"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_2"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_2_2_1_1_1"><i><font color="Green" title="E58">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_2_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_2_2_1_1_2"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  implies <font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E3:69_1_2_2"/>
for <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>  st <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">p</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">q</font> be   <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>;<br/>

<b>assume </b><a NAME="E1:69_1_2_2_2"/><i><font color="Green" title="E57">A15</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c7">q</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_2_2_2_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_2_2_1_1"/><i><font color="Green" title="E58">A16</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_2_2_2_1_1"/><b>then </b><i><font color="Green" title="E59">A17</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:69_1_2_2_2_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_2_1_1"><i><font color="Green" title="E58">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_1_2_2_2_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">q</font>)
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_2"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_2"><i><font color="Green" title="E57">A15</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_2_2_2_1_1"><i><font color="Green" title="E59">A17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_2_2_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_2_2_2_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">q</font>)
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_2"><i><font color="Green" title="E57">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E4:69_1_2_2"/>
for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">B</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E1:69_1_2_2_3"/><i><font color="Green" title="E57">A18</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c7">B</font>
 ;<br/>

<a NAME="E2:69_1_2_2_3"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="modal_1.html#T44" target="_self" title="MODAL_1:th.44">Th44</a></i>;<br/>
<a NAME="E3:69_1_2_2_3"/><b>then </b><i><font color="Green" title="E58">A19</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_2_2_3_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_3_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_2_3_1_1"/><i><font color="Green" title="E59">A20</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_2_2_3_1_1"/><b>then </b><i><font color="Green" title="E60">A21</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:69_1_2_2_3_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_3_1_1"><i><font color="Green" title="E59">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_1_2_2_3_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font></span>)</span>)
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_2"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_3"><i><font color="Green" title="E57">A18</font></i></a>, <a class="txt" href="modal_1.html#E3:69_1_2_2_3"><i><font color="Green" title="E58">A19</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_2_2_3_1_1"><i><font color="Green" title="E60">A21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_3_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_2_3_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_2_2_3_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font></span>)</span>)
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_3"><i><font color="Green" title="E57">A18</font></i></a>, <a class="txt" href="modal_1.html#E3:69_1_2_2_3"><i><font color="Green" title="E58">A19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E5:69_1_2_2"/>
for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">B</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E1:69_1_2_2_4"/><i><font color="Green" title="E57">A22</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c7">B</font>
 ;<br/>

<a NAME="E2:69_1_2_2_4"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="modal_1.html#T45" target="_self" title="MODAL_1:th.45">Th45</a></i>;<br/>
<a NAME="E3:69_1_2_2_4"/><b>then </b><i><font color="Green" title="E58">A23</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_2_2_4_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_4_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_2_4_1_1"/><i><font color="Green" title="E59">A24</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_2_2_4_1_1"/><b>then </b><i><font color="Green" title="E60">A25</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:69_1_2_2_4_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_4_1_1"><i><font color="Green" title="E59">A24</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_1_2_2_4_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font></span>)</span>)
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_2"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_4"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="modal_1.html#E3:69_1_2_2_4"><i><font color="Green" title="E58">A23</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_2_2_4_1_1"><i><font color="Green" title="E60">A25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_4_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_2_4_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_2_2_4_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font></span>)</span>)
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_4"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="modal_1.html#E3:69_1_2_2_4"><i><font color="Green" title="E58">A23</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E6:69_1_2_2"/>
for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">C</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c8">C</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>



<b>assume </b><a NAME="E1:69_1_2_2_5"/><i><font color="Green" title="E57">A26</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c8">C</font>
 ;<br/>

<a NAME="E2:69_1_2_2_5"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></i>;<br/>
<a NAME="E3:69_1_2_2_5"/><b>then </b><i><font color="Green" title="E58">A27</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/>
<a NAME="E4:69_1_2_2_5"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_5"><i><font color="Green" title="E57">A26</font></i></a>, <a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></i>;<br/>
<a NAME="E5:69_1_2_2_5"/><b>then </b><i><font color="Green" title="E59">A28</font></i>: 
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">C</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_2_2_5_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_5_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_2_5_1_1"/><i><font color="Green" title="E60">A29</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_2_2_5_1_1"/><b>then </b><i><font color="Green" title="E61">A30</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_2_2"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><a NAME="E3:69_1_2_2_5_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_5_1_1"><i><font color="Green" title="E60">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E4:69_1_2_2_5_1_1"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">C</font></span>)</span>)
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_2"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_5"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="modal_1.html#E3:69_1_2_2_5"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="modal_1.html#E5:69_1_2_2_5"><i><font color="Green" title="E59">A28</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_2_2_5_1_1"><i><font color="Green" title="E61">A30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_2_5_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_2_5_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_2_2_5_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">C</font></span>)</span>)
 <b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_2"><i><font color="Green" title="E55">A11</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_2_2_5"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="modal_1.html#E3:69_1_2_2_5"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="modal_1.html#E5:69_1_2_2_5"><i><font color="Green" title="E59">A28</font></i></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="E3:69_1"/><i><font color="Green" title="E53">A31</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="modal_1.html#E1:69_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1"><i><font color="Green" title="E52">A3</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">A</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">A</font></span>)</span>] holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font> ) );<br/>
<a NAME="E4:69_1"/><i><font color="Green" title="E54">A32</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c3">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:69_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> 
 <b>and </b><br/><a NAME="E2:69_1_3"/><i><font color="Green" title="E55">A33</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y1</font>]
 <b>and </b><br/><a NAME="E3:69_1_3"/><i><font color="Green" title="E56">A34</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">y2</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">p</font> being    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> <b> such that </b><br/><a NAME="E5:69_1_3"/><i><font color="Green" title="E57">A35</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>and </b><br/><a NAME="E6:69_1_3"/><i><font color="Green" title="E58">A36</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span>] holds <br/><font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_3"><i><font color="Green" title="E55">A33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E8:69_1_3"/><i><font color="Green" title="E59">A37</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">F</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c2">y1</font> = 
<font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>
<b>by </b><i><a class="txt" href="modal_1.html#E6:69_1_3"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="modal_1.html#E8:69_1_3"><i><font color="Green" title="E59">A37</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">y2</font>
<b>by </b><i><a class="txt" href="modal_1.html#E3:69_1_3"><i><font color="Green" title="E56">A34</font></i></a>, <a class="txt" href="modal_1.html#E5:69_1_3"><i><font color="Green" title="E57">A35</font></i></a>, <a class="txt" href="modal_1.html#E8:69_1_3"><i><font color="Green" title="E59">A37</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:69_1"/><i><font color="Green" title="E55">A38</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:69_1_4"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">x'</font> = <font color="Maroon" title="c1">x</font> as    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c3">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:69_1_4"/><i><font color="Green" title="E56">A39</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">F</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">x'</font></span>)</span>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c2">x'</font>
;<br/>

<b>thus </b><a NAME="E5:69_1_4"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x'</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">G</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E6:69_1_4"/><i><font color="Green" title="E57">A40</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">x'</font></span>)</span>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ] <b>means </b>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> $1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">x'</font></span>)</span> implies <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 );<br/>
<a NAME="E7:69_1_4"/><i><font color="Green" title="E58">A41</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_4_1"/><i><font color="Green" title="E59">A42</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">x'</font></span>)</span>
 ;<br/>

<b>hence </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  = 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i><a class="txt" href="modal_1.html#E4:69_1_4"><i><font color="Green" title="E56">A39</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a> 
<b>by </b><i><a class="txt" href="modal_1.html#E6:69_1_4"><i><font color="Green" title="E57">A40</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_4_1"><i><font color="Green" title="E59">A42</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E8:69_1_4"/><i><font color="Green" title="E59">A43</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> holds  <font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>;<br/>

<b>assume </b><a NAME="E1:69_1_4_2"/><i><font color="Green" title="E60">A44</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">x'</font></span>)</span>
 ;<br/>

<b>hence </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c5">p</font></span>)</span> = 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">p</font>)
<b>by </b><i><a class="txt" href="modal_1.html#E4:69_1_4"><i><font color="Green" title="E56">A39</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c5">p</font></span>)</span>
<b>by </b><i><a class="txt" href="modal_1.html#E6:69_1_4"><i><font color="Green" title="E57">A40</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_4_2"><i><font color="Green" title="E60">A44</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E9:69_1_4"/><i><font color="Green" title="E60">A45</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E1:69_1_4_3"/><i><font color="Green" title="E61">A46</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">A</font>]
 ;<br/>

<b>assume </b><a NAME="E2:69_1_4_3"/><i><font color="Green" title="E62">A47</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">x'</font></span>)</span>
 ;<br/>

<a NAME="E3:69_1_4_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T44" target="_self" title="MODAL_1:th.44">Th44</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span> = 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font></span>)</span>)
<b>by </b><i><a class="txt" href="modal_1.html#E4:69_1_4"><i><font color="Green" title="E56">A39</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_4_3"><i><font color="Green" title="E61">A46</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_4_3"><i><font color="Green" title="E62">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="txt" href="modal_1.html#E6:69_1_4"><i><font color="Green" title="E57">A40</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_4_3"><i><font color="Green" title="E62">A47</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E10:69_1_4"/><i><font color="Green" title="E61">A48</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>

<b>assume </b><a NAME="E1:69_1_4_4"/><i><font color="Green" title="E62">A49</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">A</font>]
 ;<br/>

<b>assume </b><a NAME="E2:69_1_4_4"/><i><font color="Green" title="E63">A50</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">x'</font></span>)</span>
 ;<br/>

<a NAME="E3:69_1_4_4"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T45" target="_self" title="MODAL_1:th.45">Th45</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font></span>)</span>)
<b>by </b><i><a class="txt" href="modal_1.html#E4:69_1_4"><i><font color="Green" title="E56">A39</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_4_4"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_4_4"><i><font color="Green" title="E63">A50</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span>
<b>by </b><i><a class="txt" href="modal_1.html#E6:69_1_4"><i><font color="Green" title="E57">A40</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_4_4"><i><font color="Green" title="E63">A50</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E11:69_1_4"/><i><font color="Green" title="E62">A51</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">A</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:69_1_4_5"/><i><font color="Green" title="E63">A52</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">A</font>]
 <b>and </b><br/><a NAME="E2:69_1_4_5"/><i><font color="Green" title="E64">A53</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c6">B</font>]
 <b>and </b><br/><a NAME="E3:69_1_4_5"/><i><font color="Green" title="E65">A54</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">x'</font></span>)</span>
 ;<br/>

<a NAME="E4:69_1_4_5"/><i><font color="Green" title="E66">A55</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></i>;<br/>
<a NAME="E5:69_1_4_5"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span> = 
<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font></span>)</span>)
<b>by </b><i><a class="txt" href="modal_1.html#E4:69_1_4"><i><font color="Green" title="E56">A39</font></i></a>, <a class="txt" href="modal_1.html#E1:69_1_4_5"><i><font color="Green" title="E63">A52</font></i></a>, <a class="txt" href="modal_1.html#E2:69_1_4_5"><i><font color="Green" title="E64">A53</font></i></a>, <a class="txt" href="modal_1.html#E3:69_1_4_5"><i><font color="Green" title="E65">A54</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_4_5"><i><font color="Green" title="E66">A55</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="txt" href="modal_1.html#E6:69_1_4"><i><font color="Green" title="E57">A40</font></i></a>, <a class="txt" href="modal_1.html#E3:69_1_4_5"><i><font color="Green" title="E65">A54</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E12:69_1_4"/>
for <font color="Olive" title="b1">p</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="modal_1.html#S1" target="_self" title="MODAL_1:sch.1">MODAL_1:sch 1</a>(<a class="txt" href="modal_1.html#E7:69_1_4"><i><font color="Green" title="E58">A41</font></i></a>, <a class="txt" href="modal_1.html#E8:69_1_4"><i><font color="Green" title="E59">A43</font></i></a>, <a class="txt" href="modal_1.html#E9:69_1_4"><i><font color="Green" title="E60">A45</font></i></a>, <a class="txt" href="modal_1.html#E10:69_1_4"><i><font color="Green" title="E61">A48</font></i></a>, <a class="txt" href="modal_1.html#E11:69_1_4"><i><font color="Green" title="E62">A51</font></i></a>);<br/></i>
<b>hence </b><a NAME="E13:69_1_4"/>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x'</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:69_1"/><i><font color="Green" title="E56">A56</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> 
 <b>and </b><br/><a NAME="E8:69_1"/><i><font color="Green" title="E57">A57</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="modal_1.html#E4:69_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="modal_1.html#E5:69_1"><i><font color="Green" title="E55">A38</font></i></a>);<br/></i>
<a NAME="E9:69_1"/>
<font color="Maroon" title="c1">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_5"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_1_5_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:69_1_5_1"/><i><font color="Green" title="E58">A58</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> 
 <b>and </b><br/><a NAME="E4:69_1_5_1"/><i><font color="Green" title="E59">A59</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E7:69_1"><i><font color="Green" title="E56">A56</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">p</font> being    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> <b> such that </b><br/><a NAME="E6:69_1_5_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>and </b><br/><a NAME="E7:69_1_5_1"/><i><font color="Green" title="E60">A60</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span>] holds <br/><font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a>, <a class="txt" href="modal_1.html#E3:69_1_5_1"><i><font color="Green" title="E58">A58</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_5_1"><i><font color="Green" title="E59">A59</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E9:69_1_5_1"/><i><font color="Green" title="E61">A61</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font></span>)</span>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<a NAME="E10:69_1_5_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E7:69_1_5_1"><i><font color="Green" title="E60">A60</font></i></a>, <a class="txt" href="modal_1.html#E9:69_1_5_1"><i><font color="Green" title="E61">A61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:69_1_5_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:69_1_5"/>
<font color="Maroon" title="c1">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="modal_1.html#E7:69_1"><i><font color="Green" title="E56">A56</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c2">F</font> = <font color="Maroon" title="c1">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon" title="c2">F</font>
;<br/>

<b>consider </b><font color="Maroon" title="c3">A</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><b> such that </b><br/><a NAME="E12:69_1"/><i><font color="Green" title="E58">A62</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">A</font> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E14:69_1"/><i><font color="Green" title="E59">A63</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<a NAME="E15:69_1"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a> 
 
<b>by </b><i><a class="txt" href="modal_1.html#E12:69_1"><i><font color="Green" title="E58">A62</font></i></a>, <a class="txt" href="modal_1.html#E14:69_1"><i><font color="Green" title="E59">A63</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:69_1"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="modal_1.html#E12:69_1"><i><font color="Green" title="E58">A62</font></i></a>, <a class="txt" href="modal_1.html#E14:69_1"><i><font color="Green" title="E59">A63</font></i></a></i>;<br/>

<b>thus </b><a NAME="E17:69_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a> holds  <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">p</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>;<br/>

<b>consider </b><font color="Maroon" title="c6">A</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><b> such that </b><br/><a NAME="E2:69_1_6"/><i><font color="Green" title="E60">A64</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c5">p</font> &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">A</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">A</font> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:69_1_6"/><i><font color="Green" title="E61">A65</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">A</font></span>)</span>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<a NAME="E5:69_1_6"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c5">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_6"><i><font color="Green" title="E60">A64</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_6"><i><font color="Green" title="E61">A65</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:69_1_6"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">p</font>)
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_6"><i><font color="Green" title="E60">A64</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_6"><i><font color="Green" title="E61">A65</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E18:69_1"/>
for <font color="Olive" title="b1">A</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  holds  <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ;<br/>

<b>consider </b><font color="Maroon" title="c6">A1</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><b> such that </b><br/><a NAME="E2:69_1_7"/><i><font color="Green" title="E60">A66</font></i>: 
( <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font> &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">A1</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">A1</font> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:69_1_7"/><i><font color="Green" title="E61">A67</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">A1</font></span>)</span>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<a NAME="E5:69_1_7"/><i><font color="Green" title="E62">A68</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>,<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:69_1_7_1"/><i><font color="Green" title="E63">A69</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">a</font> be    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:69_1_7_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 ;<br/>

<a NAME="E3:69_1_7_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_7_1"><i><font color="Green" title="E63">A69</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:69_1_7_1"/>
( ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  implies <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>  st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">C</font></span>)</span>) ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_7"><i><font color="Green" title="E60">A66</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_7"><i><font color="Green" title="E61">A67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:69_1_7"/><i><font color="Green" title="E63">A70</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_7"><i><font color="Green" title="E60">A66</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_7"><i><font color="Green" title="E61">A67</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">k</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span>;<br/>
<a NAME="E7:69_1_7"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T44" target="_self" title="MODAL_1:th.44">Th44</a></i>;<br/>
<a NAME="E8:69_1_7"/><b>then </b><i><font color="Green" title="E64">A71</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_7"><i><font color="Green" title="E62">A68</font></i></a></i>;<br/>
<a NAME="E9:69_1_7"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">B</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font> ) )
 
<b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/>
<a NAME="E10:69_1_7"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E8:69_1_7"><i><font color="Green" title="E64">A71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:69_1_7"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font></span>)</span>)
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_7"><i><font color="Green" title="E60">A66</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_7"><i><font color="Green" title="E61">A67</font></i></a>, <a class="txt" href="modal_1.html#E6:69_1_7"><i><font color="Green" title="E63">A70</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E19:69_1"/>
for <font color="Olive" title="b1">A</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  holds  <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ;<br/>

<b>consider </b><font color="Maroon" title="c6">A1</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><b> such that </b><br/><a NAME="E2:69_1_8"/><i><font color="Green" title="E60">A72</font></i>: 
( <font color="Maroon" title="c6">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font> &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">A1</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">A1</font> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:69_1_8"/><i><font color="Green" title="E61">A73</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">A1</font></span>)</span>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<a NAME="E5:69_1_8"/><i><font color="Green" title="E62">A74</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>,<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:69_1_8_1"/><i><font color="Green" title="E63">A75</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">a</font> be    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:69_1_8_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 ;<br/>

<a NAME="E3:69_1_8_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_8_1"><i><font color="Green" title="E63">A75</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:69_1_8_1"/>
( ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  implies <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>  st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">C</font></span>)</span>) ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_8"><i><font color="Green" title="E60">A72</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_8"><i><font color="Green" title="E61">A73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:69_1_8"/><i><font color="Green" title="E63">A76</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_8"><i><font color="Green" title="E60">A72</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_8"><i><font color="Green" title="E61">A73</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">k</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span>;<br/>
<a NAME="E7:69_1_8"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T45" target="_self" title="MODAL_1:th.45">Th45</a></i>;<br/>
<a NAME="E8:69_1_8"/><b>then </b><i><font color="Green" title="E64">A77</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_8"><i><font color="Green" title="E62">A74</font></i></a></i>;<br/>
<a NAME="E9:69_1_8"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">B</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font> ) )
 
<b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/>
<a NAME="E10:69_1_8"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E8:69_1_8"><i><font color="Green" title="E64">A77</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:69_1_8"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font></span>)</span>)
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_8"><i><font color="Green" title="E60">A72</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_8"><i><font color="Green" title="E61">A73</font></i></a>, <a class="txt" href="modal_1.html#E6:69_1_8"><i><font color="Green" title="E63">A76</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E20:69_1"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a>  holds  <font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">B</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ;<br/>

<b>consider </b><font color="Maroon" title="c7">A1</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><b> such that </b><br/><a NAME="E2:69_1_9"/><i><font color="Green" title="E60">A78</font></i>: 
( <font color="Maroon" title="c7">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font> &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">A1</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A1</font> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:69_1_9"/><i><font color="Green" title="E61">A79</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">A1</font></span>)</span>]
 <b>by </b><i><a class="txt" href="modal_1.html#E3:69_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/>
<a NAME="E5:69_1_9"/><i><font color="Green" title="E62">A80</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">G</font>,<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:69_1_9_1"/><i><font color="Green" title="E63">A81</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c10">a</font> be    <a href="modal_1.html#M2" title="MODAL_1:mode.2">Element</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:69_1_9_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 ;<br/>

<a NAME="E3:69_1_9_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E1:69_1_9_1"><i><font color="Green" title="E63">A81</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:69_1_9_1"/>
( ( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K17" title="MODAL_1:func.17">VERUM</a>  implies <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>  st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">p</font>) ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>  st <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">C</font></span>)</span>) ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_9"><i><font color="Green" title="E60">A78</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_9"><i><font color="Green" title="E61">A79</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:69_1_9"/><i><font color="Green" title="E63">A82</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_9"><i><font color="Green" title="E60">A78</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_9"><i><font color="Green" title="E61">A79</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">k1</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">k2</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">B</font></span>)</span>;<br/>
<a NAME="E7:69_1_9"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></i>;<br/>
<a NAME="E8:69_1_9"/><b>then </b><i><font color="Green" title="E64">A83</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_9"><i><font color="Green" title="E62">A80</font></i></a></i>;<br/>
<a NAME="E9:69_1_9"/>
 ex <font color="Olive" title="b1">B1</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <br/>( <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">B1</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B1</font> ) )
 
<b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/>
<a NAME="E10:69_1_9"/><b>then </b><i><font color="Green" title="E65">A84</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E8:69_1_9"><i><font color="Green" title="E64">A83</font></i></a></i>;<br/>
<a NAME="E11:69_1_9"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></i>;<br/>
<a NAME="E12:69_1_9"/><b>then </b><i><font color="Green" title="E66">A85</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">G</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">B</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:69_1_9"><i><font color="Green" title="E62">A80</font></i></a></i>;<br/>
<a NAME="E13:69_1_9"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="modal_1.html#K6" title="MODAL_1:func.6">MP-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">g</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">C</font></span>)</span>] holds <br/><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">C</font> ) )
 
<b>by </b><i><a class="txt" href="modal_1.html#E8:69_1"><i><font color="Green" title="E57">A57</font></i></a></i>;<br/>
<a NAME="E14:69_1_9"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E12:69_1_9"><i><font color="Green" title="E66">A85</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:69_1_9"/>
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="modal_1.html#K12" title="MODAL_1:func.12">'&amp;'</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font></span>)</span>)
 <b>by </b><i><a class="txt" href="modal_1.html#E2:69_1_9"><i><font color="Green" title="E60">A78</font></i></a>, <a class="txt" href="modal_1.html#E4:69_1_9"><i><font color="Green" title="E61">A79</font></i></a>, <a class="txt" href="modal_1.html#E6:69_1_9"><i><font color="Green" title="E63">A82</font></i></a>, <a class="txt" href="modal_1.html#E10:69_1_9"><i><font color="Green" title="E65">A84</font></i></a></i>;<br/>


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


</div>
