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

<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font> be   <a href="card_3.html#NM1" title="CARD_3:NM.1">Cardinal-Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">G</font> &amp; ( 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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) implies  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K7" title="CARD_3:func.7">Product</a> <font color="Maroon" title="c2">G</font> )</span><br/>



<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E25">A1</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">G</font> &amp; ( 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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K7" title="CARD_3:func.7">Product</a> <font color="Maroon" title="c2">G</font></span><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:52"/><i><font color="Green" title="E26">A2</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> &amp; ( 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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">Fx</font> = <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c6">Gx</font> = <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="card_3.html#E1:52"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="card_3.html#E3:52"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_3.html#E3:52_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="card_3.html#D1" target="_self" title="CARD_3:def.1">Def1</a></i>;<br/><a NAME="E5:52_1"/>
( <font color="Maroon" title="c5">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Gx</font> &amp; not <font color="Maroon" title="c5">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Fx</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E1:52"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="card_3.html#E3:52"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_3.html#E3:52_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/><a NAME="E6:52_1"/><b>then </b>
( <font color="Maroon" title="c5">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Gx</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">Fx</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Gx</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">Fx</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E3:52"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_3.html#E3:52_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:52_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E5:52"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_3.html#T37" target="_self" title="CARD_3:th.37">Th37</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:52"/><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> &amp; ( 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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="card_3.html#E5:52"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="card_3.html#D5" target="_self" title="CARD_3:def.5">Def5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>] <b>means </b>(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> &amp; ( 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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> holds <br/>( ( $1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> implies $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  ) &amp; ( $1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> implies $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) ) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon">S<sub>2</sub></font>[$1,<font color="Olive" title="b1">g</font>] );<br/>
<a NAME="E9:52"/><i><font color="Green" title="E29">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> 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>3</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="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> 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>3</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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>3</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ;<br/>
<b>consider </b><font color="Maroon" title="c7">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:52_2"/><i><font color="Green" title="E30">A10</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> &amp; ( for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> holds <br/>( ( <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">u</font>] implies <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">u</font>) ) &amp; (  not <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">u</font>] implies <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">u</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">g</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c8">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">y</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c7">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">g</font>] )</span><br/>

<b>thus </b><a NAME="E5:52_2"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">g</font>] )
 <b>by </b><i><a class="txt" href="card_3.html#E3:52_2"><i><font color="Green" title="E30">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E11:52"/><i><font color="Green" title="E30">A11</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; ( 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="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="card_3.html#E9:52"><i><font color="Green" title="E29">A9</font></i></a>);<br/></i>
<a NAME="E12:52"/><i><font color="Green" title="E31">A12</font></i>: 
<font color="Maroon" title="c6">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f1</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f1</font> or  not <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f1</font> or  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f1</font> or  not <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )</span><br/>



<b>assume </b><a NAME="E1:52_3"/><i><font color="Green" title="E32">A13</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f1</font> &amp; <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c9">g1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:52_3"/><i><font color="Green" title="E33">A14</font></i>: 
( <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g1</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">g1</font>] )
 <b>by </b><i><a class="txt" href="card_3.html#E11:52"><i><font color="Green" title="E30">A11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">g2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:52_3"/><i><font color="Green" title="E34">A15</font></i>: 
( <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c10">g2</font>] )
 <b>by </b><i><a class="txt" href="card_3.html#E11:52"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="card_3.html#E1:52_3"><i><font color="Green" title="E32">A13</font></i></a></i>;<br/>
<a NAME="E6:52_3"/><i><font color="Green" title="E35">A16</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp; <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="card_3.html#E11:52"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="card_3.html#E1:52_3"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="card_3.html#T33" target="_self" title="CARD_3:th.33">Th33</a></i>;<br/>
<a NAME="E7:52_3"/>
(  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="card_3.html#E11:52"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="card_3.html#E1:52_3"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="card_3.html#T32" target="_self" title="CARD_3:th.32">Th32</a></i>;<br/>
<a NAME="E8:52_3"/><b>then </b><i><font color="Green" title="E36">A17</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></i>;<br/>
<div><i><font color="Green" title="E37">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:52_3_1"/><b>then </b>
( <font color="Maroon" title="c9">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c10">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp; <font color="Maroon" title="c10">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c9">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="card_3.html#E3:52"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_3.html#E8:52"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="card_3.html#E1:52_3"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="card_3.html#E3:52_3"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="card_3.html#E5:52_3"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="card_3.html#E6:52_3"><i><font color="Green" title="E35">A16</font></i></a>, <a class="txt" href="card_3.html#E8:52_3"><i><font color="Green" title="E36">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E3:52_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_3.html#E1:52_3"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="card_3.html#E3:52_3"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="card_3.html#E5:52_3"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="card_3.html#E6:52_3"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E10:52_3"/>
( <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  implies ( <font color="Maroon" title="c9">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c10">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  ) )
 
<b>by </b><i><a class="txt" href="card_3.html#E3:52_3"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="card_3.html#E5:52_3"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="card_3.html#E6:52_3"><i><font color="Green" title="E35">A16</font></i></a></i>;<br/>
<a NAME="E11:52_3"/><b>then </b>
( <font color="Maroon" title="c9">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c9">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="card_3.html#E3:52"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_3.html#E8:52"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="card_3.html#E1:52_3"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="card_3.html#E3:52_3"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="card_3.html#E5:52_3"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="card_3.html#E6:52_3"><i><font color="Green" title="E35">A16</font></i></a>, <a class="txt" href="card_3.html#E9:52_3"><i><font color="Green" title="E37">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:52_3"/>
contradiction
 <b>by </b><i><a class="txt" href="card_3.html#E6:52_3"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:52"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</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="c6">f1</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">G</font> )</span><br/>

<b>assume </b><a NAME="E1:52_4"/>
<font color="Maroon" title="c7">x</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="c6">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">G</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_4"/><i><font color="Green" title="E32">A19</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:52_4"/><i><font color="Green" title="E33">A20</font></i>: 
( <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">g</font>] )
 <b>by </b><i><a class="txt" href="card_3.html#E11:52"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="card_3.html#E3:52_4"><i><font color="Green" title="E32">A19</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">G</font> implies <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/><b>assume </b><a NAME="E1:52_4_1"/><i><font color="Green" title="E34">A21</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c11">Gx</font> = <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c12">Fx</font> = <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="card_3.html#E1:52"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="card_3.html#D1" target="_self" title="CARD_3:def.1">Def1</a></i>;<br/><a NAME="E3:52_4_1"/>
<font color="Maroon" title="c12">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Gx</font>
 <b>by </b><i><a class="txt" href="card_3.html#E1:52"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="card_3.html#E1:52_4_1"><i><font color="Green" title="E34">A21</font></i></a></i>;<br/><a NAME="E4:52_4_1"/><b>then </b>
( ( <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> implies <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  ) &amp; ( <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">x</font> implies <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> ) &amp; <font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Gx</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c12">Fx</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp; <font color="Maroon" title="c12">Fx</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">Gx</font> &amp; <font color="Maroon" title="c12">Fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Fx</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E1:52"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="card_3.html#E3:52"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_3.html#E8:52"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="card_3.html#E11:52"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="card_3.html#E3:52_4"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="card_3.html#E5:52_4"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="card_3.html#E1:52_4_1"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="card_3.html#T33" target="_self" title="CARD_3:th.33">Th33</a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/><b>hence </b><a NAME="E5:52_4_1"/>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:52_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="card_3.html#E1:52"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="card_3.html#E3:52_4"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="card_3.html#E5:52_4"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="card_3.html#D5" target="_self" title="CARD_3:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:52"/><b>then </b><i><font color="Green" title="E32">A22</font></i>: 
 <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <font color="Maroon" title="c1">F</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_3.html#K7" title="CARD_3:func.7">Product</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E11:52"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="card_3.html#E12:52"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="card_1.html#T26" title="CARD_1:th.26">CARD_1:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_5"/>
 <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K7" title="CARD_3:func.7">Product</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:52_5"/><b>then </b>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span>, <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">G</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:52_5"/><i><font color="Green" title="E33">A23</font></i>: 
( <font color="Maroon" title="c7">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span>)</span>,$1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c8">K</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:52_5"/><i><font color="Green" title="E34">A24</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> &amp; ( 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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_5_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="c8">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_5_1"/><i><font color="Green" title="E35">A25</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E6:52_5"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:52_5_1"/><i><font color="Green" title="E36">A26</font></i>: 
<font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="card_3.html#E6:52_5"><i><font color="Green" title="E34">A24</font></i></a>, <a class="txt" href="card_3.html#E3:52_5_1"><i><font color="Green" title="E35">A25</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">Fx</font> = <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c11">Gx</font> = <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="card_3.html#E1:52"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="card_3.html#E3:52_5_1"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="card_3.html#D1" target="_self" title="CARD_3:def.1">Def1</a></i>;<br/><a NAME="E6:52_5_1"/><i><font color="Green" title="E37">A27</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">x</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_3.html#T31" target="_self" title="CARD_3:th.31">Th31</a>, <a class="ref" href="card_2.html#T3" title="CARD_2:th.3">CARD_2:3</a></i>;<br/><a NAME="E7:52_5_1"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Fx</font> &amp; <font color="Maroon" title="c10">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Gx</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E1:52"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="card_3.html#E3:52_5_1"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="card_3.html#T40" target="_self" title="CARD_3:th.40">Th40</a></i>;<br/><a NAME="E8:52_5_1"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">x</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">Fx</font> &amp; <font color="Maroon" title="c10">Fx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Gx</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">Gx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Gx</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E6:52_5_1"><i><font color="Green" title="E37">A27</font></i></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E9:52_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_3.html#E3:52_5_1"><i><font color="Green" title="E35">A25</font></i></a>, <a class="txt" href="card_3.html#E4:52_5_1"><i><font color="Green" title="E36">A26</font></i></a>, <a class="ref" href="card_2.html#T4" title="CARD_2:th.4">CARD_2:4</a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E8:52_5"/><b>then </b><i><font color="Green" title="E35">A28</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="card_3.html#T37" target="_self" title="CARD_3:th.37">Th37</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">K</font>;<br/><b>consider </b><font color="Maroon" title="c10">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E11:52_5"/><i><font color="Green" title="E36">A29</font></i>: 
( <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">K</font> &amp; ( 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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">K</font> holds <br/><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="card_3.html#E8:52_5"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="card_3.html#D5" target="_self" title="CARD_3:def.5">Def5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">K</font> implies <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )</span><br/><b>assume </b><a NAME="E1:52_5_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span><br/><a NAME="E2:52_5_2"/><b>then </b>
( <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> &amp; <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E6:52_5"><i><font color="Green" title="E34">A24</font></i></a></i>;<br/><b>hence </b><a NAME="E3:52_5_2"/>
<font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E13:52_5"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="card_3.html#E1:52"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="card_3.html#E6:52_5"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="card_3.html#T38" target="_self" title="CARD_3:th.38">Th38</a></i>;<br/><a NAME="E14:52_5"/><b>then </b>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="card_3.html#E8:52_5"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:52_5"/><i><font color="Green" title="E37">A30</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E4:52_5"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:52_5"/><i><font color="Green" title="E38">A31</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> &amp; <font color="Maroon" title="c12">X</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> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_3.html#E4:52_5"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="card_3.html#E16:52_5"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:52_5"/><i><font color="Green" title="E39">A32</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E18:52_5"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E21:52_5"/>
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c12">X</font>
 <b>by </b><i><a class="txt" href="card_3.html#E11:52_5"><i><font color="Green" title="E36">A29</font></i></a>, <a class="txt" href="card_3.html#E16:52_5"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="card_3.html#E18:52_5"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E22:52_5"/><b>then </b>
( <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">F</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E20:52_5"><i><font color="Green" title="E39">A32</font></i></a>, <a class="ref" href="card_3.html#D3" target="_self" title="CARD_3:def.3">Def3</a>, <a class="ref" href="card_3.html#D6" target="_self" title="CARD_3:def.6">Def6</a></i>;<br/><a NAME="E23:52_5"/><b>then </b>
( not <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">x</font></span>)</span> &amp; <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="txt" href="card_3.html#E6:52_5"><i><font color="Green" title="E34">A24</font></i></a>, <a class="txt" href="card_3.html#E11:52_5"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E24:52_5"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:52"/>
 <a href="card_3.html#K6" title="CARD_3:func.6">Sum</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K7" title="CARD_3:func.7">Product</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="card_3.html#E14:52"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
