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

<span class="kw">defpred </span><font color="Maroon">S<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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</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> ] <span class="kw">means </span>( ( $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/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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>1</sub></font>[<font color="Olive" title="b1">F</font>,$1];<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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>1</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="E1:75_1"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
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="75_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c1">k</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">given </span><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>()<span class="kw"> such that </span><a NAME="E2:75_1_1"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">defpred </span><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> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()] <span class="kw">means </span>( (  <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>2</sub></font>[$2,<font color="Maroon" title="c2">F</font>,$1] ) );<br/>
<a NAME="E3:75_1_1"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
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>5</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="75_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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 <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_1_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 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><span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T48" target="_self" title="MODAL_1:th.48">Th48</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:75_1_1_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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 verum</span><br/><div class="add"><span class="kw">take </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:75_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
(  <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c4">y</font> = <font color="Maroon">F<sub>2</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><span class="kw">thus </span><a NAME="E2:75_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="modal_1.html#E75"><span class="lab"><font color="Green" title="E48">Lm4</font></span></a>, <a class="txt" href="modal_1.html#E77"><span class="lab"><font color="Green" title="E50">Lm6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:75_1_1_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">p</font> being   <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a><span class="kw"> such that </span><br/><a NAME="E3:75_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
<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/><span class="kw">take </span><font color="Maroon" title="c5">y</font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">p</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><span class="kw">thus </span><a NAME="E4:75_1_1_1_1_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="modal_1.html#E77"><span class="lab"><font color="Green" title="E50">Lm6</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:75_1_1_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">B</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><span class="kw"> such that </span><br/><a NAME="E3:75_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<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/><span class="kw">take </span><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>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><span class="kw">thus </span><a NAME="E4:75_1_1_1_1_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="txt" href="modal_1.html#E75"><span class="lab"><font color="Green" title="E48">Lm4</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1_5"><span class="kw">case </span></a><a NAME="E1:75_1_1_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">B</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><span class="kw"> such that </span><br/><a NAME="E3:75_1_1_1_1_1_5"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
<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/><span class="kw">take </span><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>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><span class="kw">thus </span><a NAME="E4:75_1_1_1_1_1_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="txt" href="modal_1.html#E75"><span class="lab"><font color="Green" title="E48">Lm4</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1_6"><span class="kw">case </span></a><a NAME="E1:75_1_1_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:75_1_1_1_1_1_6"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
<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/><span class="kw">take </span><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>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_1_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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>) )</span><br/><span class="kw">assume </span><a NAME="E1:75_1_1_1_1_1_6_1"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><a NAME="E2:75_1_1_1_1_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="ref" href="modal_1.html#T40" target="_self" title="MODAL_1:th.40">Th40</a></span>;<br/><span class="kw">hence </span><a NAME="E3:75_1_1_1_1_1_6_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E1:75_1_1_1_1_1_6_1"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="modal_1.html#T40" target="_self" title="MODAL_1:th.40">Th40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:75_1_1_1_1_1_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="modal_1.html#E75"><span class="lab"><font color="Green" title="E48">Lm4</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:75_1_1_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 <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E5:75_1_1"/><span class="lab"><font color="Green" title="E54">A10</font></span>: 
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>,<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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E3:75_1_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>);<br/></span>
<span class="kw">take </span><font color="Maroon" title="c4">H</font> = <font color="Maroon" title="c3">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</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]</span><br/>

<span class="kw">thus </span><a NAME="E6:75_1_1"/>
<font color="Maroon">S<sub>1</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]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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 implies ( ( <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>() ) &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="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>) ) &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="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>) ) &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="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>) ) &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="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>) ) ) )</span><br/>

<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:75_1_1_2"/><span class="lab"><font color="Green" title="E55">A11</font></span>: 
 <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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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>() ) &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="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>) ) &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="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>) ) &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="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>) ) &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="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>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:75_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>() )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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>) ) &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="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>) ) &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="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>) ) &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="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>) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_1_1_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="75_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_1_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="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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>() )</span><br/><div class="add"><a NAME="E2:75_1_1_2_1_1_1"/><span class="kw">then </span>
(  <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> &amp; <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:75_1_1_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>() )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>() )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:75_1_1_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>() )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:75_1_1_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>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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>) ) &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="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>) ) &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="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>) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">q</font> be   <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> 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>3</sub></font>(<font color="Maroon" title="c7">q</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_1_2_2"/><span class="lab"><font color="Green" title="E56">A12</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_1_1_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="75_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_1_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><div class="add"><a NAME="E2:75_1_1_2_2_1_1"/><span class="kw">then </span>
(  <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> &amp; <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:75_1_1_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_2_2"><span class="lab"><font color="Green" title="E56">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:75_1_1_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E1:75_1_1_2_2"><span class="lab"><font color="Green" title="E56">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:75_1_1_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>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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>) ) &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="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>) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">B</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> 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>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>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_1_2_3"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<a NAME="E2:75_1_1_2_3"/><span class="kw">then </span>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="modal_1.html#T44" target="_self" title="MODAL_1:th.44">Th44</a></span>;<br/>
<a NAME="E3:75_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A14</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_1_1_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="75_1_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_2_3_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><div class="add"><a NAME="E2:75_1_1_2_3_1_1"/><span class="kw">then </span>
(  <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> &amp; <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:75_1_1_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_2_3"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E3:75_1_1_2_3"><span class="lab"><font color="Green" title="E57">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:75_1_1_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E1:75_1_1_2_3"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E3:75_1_1_2_3"><span class="lab"><font color="Green" title="E57">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:75_1_1_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>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">B</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> 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>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>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_1_2_4"/><span class="lab"><font color="Green" title="E56">A15</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<a NAME="E2:75_1_1_2_4"/><span class="kw">then </span>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="modal_1.html#T45" target="_self" title="MODAL_1:th.45">Th45</a></span>;<br/>
<a NAME="E3:75_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A16</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_1_1_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="75_1_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_2_4_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><div class="add"><a NAME="E2:75_1_1_2_4_1_1"/><span class="kw">then </span>
(  <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> &amp; <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:75_1_1_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_2_4"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E3:75_1_1_2_4"><span class="lab"><font color="Green" title="E57">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:75_1_1_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E1:75_1_1_2_4"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E3:75_1_1_2_4"><span class="lab"><font color="Green" title="E57">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:75_1_1_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>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> 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>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>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:75_1_1_2_5"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<a NAME="E2:75_1_1_2_5"/><span class="kw">then </span>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></span>;<br/>
<a NAME="E3:75_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A18</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a></span>;<br/>
<a NAME="E4:75_1_1_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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E1:75_1_1_2_5"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></span>;<br/>
<a NAME="E5:75_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A19</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_1_1_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="75_1_1_2_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_2_5_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><div class="add"><a NAME="E2:75_1_1_2_5_1_1"/><span class="kw">then </span>
(  <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> &amp; <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E1:75_1_1_2"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:75_1_1_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_2_5"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E3:75_1_1_2_5"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="txt" href="#E5:75_1_1_2_5"><span class="lab"><font color="Green" title="E58">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:75_1_1_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E1:75_1_1_2_5"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E3:75_1_1_2_5"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="txt" href="#E5:75_1_1_2_5"><span class="lab"><font color="Green" title="E58">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E2:75_1"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
<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="75_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><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/>
<span class="kw">take </span>
<font color="Maroon" title="c1">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">F</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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>  implies ( ( <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>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:75_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="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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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>) ) )</span><br/>

<span class="kw">hence </span><a NAME="E3:75_1_2"/>
( ( <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>) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:75_1"/><span class="lab"><font color="Green" title="E53">A21</font></span>: 
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:75_1"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>);<br/></span>
<a NAME="E4:75_1"/><span class="lab"><font color="Green" title="E54">A22</font></span>: 
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="75_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  implies  ex <font color="Olive" title="b1">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="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c1">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">x9</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/>
<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E4:75_1_3"/><span class="lab"><font color="Green" title="E55">A23</font></span>: 
<font color="Maroon">S<sub>1</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">x9</font></span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E53">A21</font></span></a></span>;<br/>
<span class="kw">take </span>
<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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c1">x</font>,<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>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">x9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</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>1</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="c2">x9</font></span>)</span>] holds <br/><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="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x9</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:75_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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>1</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="c2">x9</font></span>)</span>] holds <br/><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="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x9</font></span><br/>

<span class="kw">let </span><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>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</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">x9</font></span>)</span>] implies <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">x9</font> )</span><br/>

<span class="kw">defpred </span><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> ] <span class="kw">means </span>(  <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">x9</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/>
<span class="kw">assume </span><a NAME="E6:75_1_3"/><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<font color="Maroon">S<sub>1</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">x9</font></span>)</span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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">x9</font></span><br/>

<a NAME="E7:75_1_3"/><span class="lab"><font color="Green" title="E57">A25</font></span>: 
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="75_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K16" title="MODAL_1:func.16">@</a> <font color="Maroon" title="c5">p</font>]</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_3_1"/><span class="lab"><font color="Green" title="E58">A26</font></span>: 
 <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">x9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <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> <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></span><br/>

<span class="kw">hence </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75_1_3"><span class="lab"><font color="Green" title="E55">A23</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_3"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="txt" href="#E1:75_1_3_1"><span class="lab"><font color="Green" title="E58">A26</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:75_1_3"/><span class="lab"><font color="Green" title="E58">A27</font></span>: 
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="75_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">A</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c6">B</font>] implies <font color="Maroon">S<sub>5</sub></font>[<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><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:75_1_3_2"/><span class="lab"><font color="Green" title="E59">A28</font></span>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">A</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c6">B</font>] )
 <span class="kw">and </span><br/><a NAME="E2:75_1_3_2"/><span class="lab"><font color="Green" title="E60">A29</font></span>: 
 <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">x9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <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> <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></span><br/>

<a NAME="E3:75_1_3_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="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> &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="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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></span>;<br/>
<span class="kw">hence </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75_1_3"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="txt" href="#E1:75_1_3_2"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E2:75_1_3_2"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_3"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="txt" href="#E2:75_1_3_2"><span class="lab"><font color="Green" title="E60">A29</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:75_1_3"/><span class="lab"><font color="Green" title="E59">A30</font></span>: 
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="75_1_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">A</font>] implies <font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_3_3"/><span class="lab"><font color="Green" title="E60">A31</font></span>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">A</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font>]</span><br/>

<span class="kw">assume </span><a NAME="E2:75_1_3_3"/><span class="lab"><font color="Green" title="E61">A32</font></span>: 
 <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">x9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <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> <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></span><br/>

<a NAME="E3:75_1_3_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#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T45" target="_self" title="MODAL_1:th.45">Th45</a></span>;<br/>
<span class="kw">hence </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75_1_3"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="txt" href="#E1:75_1_3_3"><span class="lab"><font color="Green" title="E60">A31</font></span></a>, <a class="txt" href="#E2:75_1_3_3"><span class="lab"><font color="Green" title="E61">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_3"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="txt" href="#E2:75_1_3_3"><span class="lab"><font color="Green" title="E61">A32</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:75_1_3"/><span class="lab"><font color="Green" title="E60">A33</font></span>: 
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="75_1_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">A</font>] implies <font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_3_4"/><span class="lab"><font color="Green" title="E61">A34</font></span>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">A</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[ <a href="modal_1.html#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font>]</span><br/>

<span class="kw">assume </span><a NAME="E2:75_1_3_4"/><span class="lab"><font color="Green" title="E62">A35</font></span>: 
 <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">x9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <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> <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></span><br/>

<a NAME="E3:75_1_3_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#K10" title="MODAL_1:func.10">'not'</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T44" target="_self" title="MODAL_1:th.44">Th44</a></span>;<br/>
<span class="kw">hence </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75_1_3"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="txt" href="#E1:75_1_3_4"><span class="lab"><font color="Green" title="E61">A34</font></span></a>, <a class="txt" href="#E2:75_1_3_4"><span class="lab"><font color="Green" title="E62">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_3"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="txt" href="#E2:75_1_3_4"><span class="lab"><font color="Green" title="E62">A35</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:75_1_3"/><span class="lab"><font color="Green" title="E61">A36</font></span>: 
<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="75_1_3_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:75_1_3_5"/><span class="lab"><font color="Green" title="E62">A37</font></span>: 
 <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">x9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>  <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> </span><br/>

<span class="kw">hence </span><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>()
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75_1_3"><span class="lab"><font color="Green" title="E55">A23</font></span></a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_3"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="txt" href="#E1:75_1_3_5"><span class="lab"><font color="Green" title="E62">A37</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:75_1_3"/>
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="modal_1.html#S1" target="_self" title="MODAL_1:sch.1">MODAL_1:sch 1</a>(<a class="txt" href="#E11:75_1_3"><span class="lab"><font color="Green" title="E61">A36</font></span></a>, <a class="txt" href="#E7:75_1_3"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="txt" href="#E10:75_1_3"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="txt" href="#E9:75_1_3"><span class="lab"><font color="Green" title="E59">A30</font></span></a>, <a class="txt" href="#E8:75_1_3"><span class="lab"><font color="Green" title="E58">A27</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E13:75_1_3"/>
<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">x9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:75_1"/><span class="lab"><font color="Green" title="E55">A38</font></span>: 
 <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> 
 <span class="kw">and </span><br/><a NAME="E7:75_1"/><span class="lab"><font color="Green" title="E56">A39</font></span>: 
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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E4:75_1"><span class="lab"><font color="Green" title="E54">A22</font></span></a>);<br/></span>
<a NAME="E8:75_1"/>
 <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="75_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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>() )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_4"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>()</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:75_1_4"/><span class="lab"><font color="Green" title="E57">A40</font></span>: 
( <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>  &amp; <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E5:75_1_4"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">and </span><br/><a NAME="E6:75_1_4"/><span class="lab"><font color="Green" title="E58">A41</font></span>: 
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>1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a>, <a class="txt" href="#E3:75_1_4"><span class="lab"><font color="Green" title="E57">A40</font></span></a></span>;<br/>
<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E8:75_1_4"/><span class="lab"><font color="Green" title="E59">A42</font></span>: 
<font color="Maroon">S<sub>1</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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E53">A21</font></span></a></span>;<br/>
<a NAME="E9:75_1_4"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_4"><span class="lab"><font color="Green" title="E58">A41</font></span></a>, <a class="txt" href="#E8:75_1_4"><span class="lab"><font color="Green" title="E59">A42</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:75_1_4"/>
<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>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1"><span class="lab"><font color="Green" title="E55">A38</font></span></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></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">A</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><span class="kw"> such that </span><br/><a NAME="E11:75_1"/><span class="lab"><font color="Green" title="E57">A43</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E12:75_1"/><span class="lab"><font color="Green" title="E58">A44</font></span>: 
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>1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>() &amp; ( 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>) ) &amp; ( 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>) ) &amp; ( 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>) ) &amp; ( 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>) ) )</span><br/>

<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E14:75_1"/><span class="lab"><font color="Green" title="E59">A45</font></span>: 
<font color="Maroon">S<sub>1</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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E53">A21</font></span></a></span>;<br/>
<a NAME="E15:75_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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:75_1"><span class="lab"><font color="Green" title="E57">A43</font></span></a>, <a class="txt" href="#E12:75_1"><span class="lab"><font color="Green" title="E58">A44</font></span></a>, <a class="txt" href="#E14:75_1"><span class="lab"><font color="Green" title="E59">A45</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:75_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>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:75_1"><span class="lab"><font color="Green" title="E57">A43</font></span></a>, <a class="txt" href="#E14:75_1"><span class="lab"><font color="Green" title="E59">A45</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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>) ) &amp; ( 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>) ) &amp; ( 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>) ) &amp; ( 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>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E17:75_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>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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>) ) &amp; ( 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>) ) &amp; ( 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>) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="modal_1.html#NM1" title="MODAL_1:NM.1">MP-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">A</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><span class="kw"> such that </span><br/><a NAME="E2:75_1_5"/><span class="lab"><font color="Green" title="E60">A46</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:75_1_5"/><span class="lab"><font color="Green" title="E61">A47</font></span>: 
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>1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/>
<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E5:75_1_5"/><span class="lab"><font color="Green" title="E62">A48</font></span>: 
<font color="Maroon">S<sub>1</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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E53">A21</font></span></a></span>;<br/>
<a NAME="E6:75_1_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_5"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="txt" href="#E3:75_1_5"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E5:75_1_5"><span class="lab"><font color="Green" title="E62">A48</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:75_1_5"/>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_5"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="txt" href="#E5:75_1_5"><span class="lab"><font color="Green" title="E62">A48</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E18:75_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>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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>) ) &amp; ( 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>) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">A1</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><span class="kw"> such that </span><br/><a NAME="E2:75_1_6"/><span class="lab"><font color="Green" title="E60">A49</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:75_1_6"/><span class="lab"><font color="Green" title="E61">A50</font></span>: 
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>1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/>
<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E5:75_1_6"/><span class="lab"><font color="Green" title="E62">A51</font></span>: 
<font color="Maroon">S<sub>1</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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E53">A21</font></span></a></span>;<br/>
<a NAME="E6:75_1_6"/><span class="lab"><font color="Green" title="E63">A52</font></span>: 
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>1</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="75_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c8">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_6_1"/><span class="lab"><font color="Green" title="E64">A53</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c8">k</font>]</span><br/>

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> implies ( ( <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>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:75_1_6_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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>) ) )</span><br/>

<a NAME="E3:75_1_6_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_6_1"><span class="lab"><font color="Green" title="E64">A53</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:75_1_6_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>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_6"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="txt" href="#E5:75_1_6"><span class="lab"><font color="Green" title="E62">A51</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:75_1_6"/><span class="lab"><font color="Green" title="E64">A54</font></span>: 
 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>1</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> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/>
<span class="kw">set </span><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="E8:75_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="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T44" target="_self" title="MODAL_1:th.44">Th44</a></span>;<br/>
<a NAME="E9:75_1_6"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</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>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_6"><span class="lab"><font color="Green" title="E63">A52</font></span></a></span>;<br/>
<a NAME="E10:75_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A55</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1_6"><span class="lab"><font color="Green" title="E64">A54</font></span></a></span>;<br/>
<a NAME="E11:75_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#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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_6"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="txt" href="#E3:75_1_6"><span class="lab"><font color="Green" title="E61">A50</font></span></a>, <a class="txt" href="#E5:75_1_6"><span class="lab"><font color="Green" title="E62">A51</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:75_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#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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_6"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="txt" href="#E5:75_1_6"><span class="lab"><font color="Green" title="E62">A51</font></span></a>, <a class="txt" href="#E10:75_1_6"><span class="lab"><font color="Green" title="E65">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E19:75_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>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">A1</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><span class="kw"> such that </span><br/><a NAME="E2:75_1_7"/><span class="lab"><font color="Green" title="E60">A56</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:75_1_7"/><span class="lab"><font color="Green" title="E61">A57</font></span>: 
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>1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/>
<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E5:75_1_7"/><span class="lab"><font color="Green" title="E62">A58</font></span>: 
<font color="Maroon">S<sub>1</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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E53">A21</font></span></a></span>;<br/>
<a NAME="E6:75_1_7"/><span class="lab"><font color="Green" title="E63">A59</font></span>: 
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>1</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="75_1_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c8">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_7_1"/><span class="lab"><font color="Green" title="E64">A60</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">G</font>,<font color="Maroon" title="c8">k</font>]</span><br/>

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> implies ( ( <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>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:75_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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>) ) )</span><br/>

<a NAME="E3:75_1_7_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_7_1"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:75_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>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_7"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E5:75_1_7"><span class="lab"><font color="Green" title="E62">A58</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:75_1_7"/><span class="lab"><font color="Green" title="E64">A61</font></span>: 
 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>1</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> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/>
<span class="kw">set </span><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="E8:75_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#K11" title="MODAL_1:func.11">(#)</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T45" target="_self" title="MODAL_1:th.45">Th45</a></span>;<br/>
<a NAME="E9:75_1_7"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</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>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_7"><span class="lab"><font color="Green" title="E63">A59</font></span></a></span>;<br/>
<a NAME="E10:75_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A62</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1_7"><span class="lab"><font color="Green" title="E64">A61</font></span></a></span>;<br/>
<a NAME="E11:75_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#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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_7"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E3:75_1_7"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="txt" href="#E5:75_1_7"><span class="lab"><font color="Green" title="E62">A58</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:75_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#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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_7"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="txt" href="#E5:75_1_7"><span class="lab"><font color="Green" title="E62">A58</font></span></a>, <a class="txt" href="#E10:75_1_7"><span class="lab"><font color="Green" title="E65">A62</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E20:75_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>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">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/>
<span class="kw">set </span><font color="Maroon" title="c8">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/>
<span class="kw">consider </span><font color="Maroon" title="c9">A1</font> being   <a href="modal_1.html#NM5" title="MODAL_1:NM.5">MP-wff</a><span class="kw"> such that </span><br/><a NAME="E2:75_1_8"/><span class="lab"><font color="Green" title="E60">A63</font></span>: 
<font color="Maroon" title="c9">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>
 <span class="kw">and </span><br/><a NAME="E3:75_1_8"/><span class="lab"><font color="Green" title="E61">A64</font></span>: 
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>1</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="c9">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="c9">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">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>()<span class="kw"> such that </span><br/><a NAME="E5:75_1_8"/><span class="lab"><font color="Green" title="E62">A65</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">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="c9">A1</font></span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E53">A21</font></span></a></span>;<br/>
<a NAME="E6:75_1_8"/><span class="lab"><font color="Green" title="E63">A66</font></span>: 
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>1</sub></font>[<font color="Maroon" title="c10">G</font>,<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">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> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">G</font>,<font color="Maroon" title="c11">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_8_1"/><span class="lab"><font color="Green" title="E64">A67</font></span>: 
<font color="Maroon" title="c11">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">G</font>,<font color="Maroon" title="c11">k</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c12">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="c12">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> implies ( ( <font color="Maroon" title="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c10">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c10">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c10">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">C</font></span>)</span>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:75_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="c12">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c10">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c10">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c10">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">C</font></span>)</span>) ) )</span><br/>

<a NAME="E3:75_1_8_1"/><span class="kw">then </span>
 <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="c12">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_8_1"><span class="lab"><font color="Green" title="E64">A67</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:75_1_8_1"/>
( ( <font color="Maroon" title="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c10">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c10">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="c12">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">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="c10">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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">C</font></span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_8"><span class="lab"><font color="Green" title="E60">A63</font></span></a>, <a class="txt" href="#E5:75_1_8"><span class="lab"><font color="Green" title="E62">A65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:75_1_8"/><span class="lab"><font color="Green" title="E64">A68</font></span>: 
 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>1</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> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/>
<a NAME="E8:75_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"><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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></span>;<br/>
<a NAME="E9:75_1_8"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">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>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_8"><span class="lab"><font color="Green" title="E63">A66</font></span></a></span>;<br/>
<a NAME="E10:75_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A69</font></span>: 
<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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1_8"><span class="lab"><font color="Green" title="E64">A68</font></span></a></span>;<br/>
<a NAME="E11:75_1_8"/><span class="lab"><font color="Green" title="E66">A70</font></span>: 
 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>1</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> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75_1"><span class="lab"><font color="Green" title="E56">A39</font></span></a></span>;<br/>
<a NAME="E12:75_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="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#T46" target="_self" title="MODAL_1:th.46">Th46</a></span>;<br/>
<a NAME="E13:75_1_8"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">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>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_8"><span class="lab"><font color="Green" title="E63">A66</font></span></a></span>;<br/>
<a NAME="E14:75_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A71</font></span>: 
<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="c10">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:75_1_8"><span class="lab"><font color="Green" title="E66">A70</font></span></a></span>;<br/>
<a NAME="E15:75_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"><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="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_8"><span class="lab"><font color="Green" title="E60">A63</font></span></a>, <a class="txt" href="#E3:75_1_8"><span class="lab"><font color="Green" title="E61">A64</font></span></a>, <a class="txt" href="#E5:75_1_8"><span class="lab"><font color="Green" title="E62">A65</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:75_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"><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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_8"><span class="lab"><font color="Green" title="E60">A63</font></span></a>, <a class="txt" href="#E5:75_1_8"><span class="lab"><font color="Green" title="E62">A65</font></span></a>, <a class="txt" href="#E10:75_1_8"><span class="lab"><font color="Green" title="E65">A69</font></span></a>, <a class="txt" href="#E14:75_1_8"><span class="lab"><font color="Green" title="E67">A71</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
