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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49"/><span class="lab"><font color="Green" title="E26">A1</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E27">A2</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="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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <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/>
<span class="kw">consider </span><font color="Maroon" title="c3">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="E4:49"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
(  <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>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:49_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:49_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E3:49_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="card_3.html#D1" target="_self" title="CARD_3:def.1">Def1</a></span>;<br/><a NAME="E6:49_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E3:49_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a></span>;<br/><a NAME="E7:49_1"/>
 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>
 ;<br/><a NAME="E8:49_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:49_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E3:49_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A7</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T26" target="_self" title="CARD_3:th.26">Th37</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">a</font> =  the    <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/>
<span class="kw">consider </span><font color="Maroon" title="c5">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:49"/>
 the    <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font>
 <span class="kw">and </span><br/><a NAME="E9:49"/>
 <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>
 <span class="kw">and </span><br/><a NAME="E10:49"/><span class="lab"><font color="Green" title="E30">A8</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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="card_3.html#D5" target="_self" title="CARD_3:def.5">Def5</a></span>;<br/>
<span class="kw">defpred </span><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>] <span class="kw">means </span>(  <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/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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="E11:49"/><span class="lab"><font color="Green" title="E31">A9</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="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="49_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:49_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:()"><span class="comment"><font color="firebrick">::  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>3</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <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/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$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/>
<span class="kw">consider </span><font color="Maroon" title="c7">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:49_2"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
(  <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>) ) ) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></span>
<span class="kw">reconsider </span><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/>
<span class="kw">take </span>
<font color="Maroon" title="c8">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">take </span>
<font color="Maroon" title="c7">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E5:49_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>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c6">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E13:49"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
(  <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>] ) )
 <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="#E11:49"><span class="lab"><font color="Green" title="E31">A9</font></span></a>);<br/></span>
<a NAME="E14:49"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<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="49_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELAT_1:func.1">proj1</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#K1" title="RELAT_1:func.1">proj1</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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELAT_1:func.1">proj1</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#K1" title="RELAT_1:func.1">proj1</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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49_3"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:49_3"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:49_3"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:49_3"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">g1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:49_3"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:49_3"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">g1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E1:49_3"><span class="lab"><font color="Green" title="E34">A13</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">g2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E9:49_3"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E10:49_3"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c10">g2</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E2:49_3"><span class="lab"><font color="Green" title="E35">A14</font></span></a></span>;<br/>
<a NAME="E11:49_3"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E1:49_3"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="card_3.html#T22" target="_self" title="CARD_3:th.22">Th33</a></span>;<br/>
<a NAME="E12:49_3"/><span class="lab"><font color="Green" title="E43">A22</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E2:49_3"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="card_3.html#T22" target="_self" title="CARD_3:th.22">Th33</a></span>;<br/>
<a NAME="E13:49_3"/><span class="lab"><font color="Green" title="E44">A23</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E2:49_3"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="card_3.html#T22" target="_self" title="CARD_3:th.22">Th33</a></span>;<br/>
<a NAME="E14:49_3"/><span class="lab"><font color="Green" title="E45">A24</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E1:49_3"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="card_3.html#T21" target="_self" title="CARD_3:th.21">Th32</a></span>;<br/>
<a NAME="E15:49_3"/><span class="lab"><font color="Green" title="E46">A25</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E2:49_3"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="card_3.html#T21" target="_self" title="CARD_3:th.21">Th32</a></span>;<br/>
<a NAME="E16:49_3"/><span class="lab"><font color="Green" title="E47">A26</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:49_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></span>;<br/>
<a NAME="E17:49_3"/><span class="lab"><font color="Green" title="E48">A27</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:49_3"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E49">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_3_1"/><span class="lab"><font color="Green" title="E50">A29</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:49_3_1"/><span class="lab"><font color="Green" title="E51">A30</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_3"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E12:49_3"><span class="lab"><font color="Green" title="E43">A22</font></span></a></span>;<br/><a NAME="E3:49_3_1"/><span class="lab"><font color="Green" title="E52">A31</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_3"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="txt" href="#E7:49_3"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E12:49_3"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E16:49_3"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="txt" href="#E17:49_3"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="txt" href="#E1:49_3_1"><span class="lab"><font color="Green" title="E50">A29</font></span></a></span>;<br/><a NAME="E4:49_3_1"/><span class="lab"><font color="Green" title="E53">A32</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E12:49_3"><span class="lab"><font color="Green" title="E43">A22</font></span></a></span>;<br/><a NAME="E5:49_3_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E10:49"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E12:49_3"><span class="lab"><font color="Green" title="E43">A22</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:49_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_3"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E6:49_3"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E9:49_3"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E13:49_3"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E2:49_3_1"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="txt" href="#E3:49_3_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="txt" href="#E4:49_3_1"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E19:49_3"/><span class="lab"><font color="Green" title="E50">A33</font></span>: 
( <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>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_3"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E10:49_3"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E11:49_3"><span class="lab"><font color="Green" title="E42">A21</font></span></a></span>;<br/>
<a NAME="E20:49_3"/><span class="lab"><font color="Green" title="E51">A34</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_3"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E6:49_3"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E9:49_3"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E10:49_3"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E12:49_3"><span class="lab"><font color="Green" title="E43">A22</font></span></a></span>;<br/>
<a NAME="E21:49_3"/><span class="lab"><font color="Green" title="E52">A35</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_3"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E6:49_3"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E7:49_3"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E9:49_3"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E12:49_3"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E18:49_3"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E19:49_3"><span class="lab"><font color="Green" title="E50">A33</font></span></a></span>;<br/>
<a NAME="E22:49_3"/><span class="lab"><font color="Green" title="E53">A36</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E12:49_3"><span class="lab"><font color="Green" title="E43">A22</font></span></a></span>;<br/>
<a NAME="E23:49_3"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E10:49"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E12:49_3"><span class="lab"><font color="Green" title="E43">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:49_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49_3"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="txt" href="#E20:49_3"><span class="lab"><font color="Green" title="E51">A34</font></span></a>, <a class="txt" href="#E21:49_3"><span class="lab"><font color="Green" title="E52">A35</font></span></a>, <a class="txt" href="#E22:49_3"><span class="lab"><font color="Green" title="E53">A36</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:49"/>
 <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="49_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:49_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:49_4"/><span class="lab"><font color="Green" title="E34">A37</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:49_4"/><span class="lab"><font color="Green" title="E35">A38</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:49_4"/><span class="lab"><font color="Green" title="E36">A39</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:49_4"/><span class="lab"><font color="Green" title="E37">A40</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">g</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E3:49_4"><span class="lab"><font color="Green" title="E34">A37</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E1:49_4_1"/><span class="lab"><font color="Green" title="E38">A41</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="card_3.html#D1" target="_self" title="CARD_3:def.1">Def1</a></span>;<br/><a NAME="E3:49_4_1"/><span class="lab"><font color="Green" title="E39">A42</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:49_4_1"><span class="lab"><font color="Green" title="E38">A41</font></span></a></span>;<br/><a NAME="E4:49_4_1"/><span class="lab"><font color="Green" title="E40">A43</font></span>: 
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E7:49_4"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="txt" href="#E1:49_4_1"><span class="lab"><font color="Green" title="E38">A41</font></span></a></span>;<br/><a NAME="E5:49_4_1"/><span class="lab"><font color="Green" title="E41">A44</font></span>: 
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E7:49_4"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="txt" href="#E1:49_4_1"><span class="lab"><font color="Green" title="E38">A41</font></span></a></span>;<br/><a NAME="E6:49_4_1"/><span class="lab"><font color="Green" title="E42">A45</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E10:49"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E1:49_4_1"><span class="lab"><font color="Green" title="E38">A41</font></span></a></span>;<br/><a NAME="E7:49_4_1"/><span class="lab"><font color="Green" title="E43">A46</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E1:49_4_1"><span class="lab"><font color="Green" title="E38">A41</font></span></a></span>;<br/><a NAME="E8:49_4_1"/><span class="lab"><font color="Green" title="E44">A47</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E3:49_4"><span class="lab"><font color="Green" title="E34">A37</font></span></a>, <a class="ref" href="card_3.html#T22" target="_self" title="CARD_3:th.22">Th33</a></span>;<br/><a NAME="E9:49_4_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_4_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="ref" href="card_1.html#T3" title="CARD_1:th.3">CARD_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:49_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_4_1"><span class="lab"><font color="Green" title="E40">A43</font></span></a>, <a class="txt" href="#E5:49_4_1"><span class="lab"><font color="Green" title="E41">A44</font></span></a>, <a class="txt" href="#E6:49_4_1"><span class="lab"><font color="Green" title="E42">A45</font></span></a>, <a class="txt" href="#E7:49_4_1"><span class="lab"><font color="Green" title="E43">A46</font></span></a>, <a class="txt" href="#E8:49_4_1"><span class="lab"><font color="Green" title="E44">A47</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:49_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E4:49_4"><span class="lab"><font color="Green" title="E35">A38</font></span></a>, <a class="txt" href="#E6:49_4"><span class="lab"><font color="Green" title="E36">A39</font></span></a>, <a class="txt" href="#E7:49_4"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="ref" href="card_3.html#D5" target="_self" title="CARD_3:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A48</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E14:49"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="card_1.html#T10" title="CARD_1:th.10">CARD_1:10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:49_5"/><span class="kw">then </span>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" title="CARD_1:th.5">CARD_1:5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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="E4:49_5"/>
<font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E5:49_5"/><span class="lab"><font color="Green" title="E35">A49</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E6:49_5"/><span class="lab"><font color="Green" title="E36">A50</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <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/><span class="kw">consider </span><font color="Maroon" title="c8">K</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:49_5"/><span class="lab"><font color="Green" title="E37">A51</font></span>: 
(  <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>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:49_5_1"/><span class="lab"><font color="Green" title="E38">A52</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:49_5_1"/><span class="lab"><font color="Green" title="E39">A53</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_5"><span class="lab"><font color="Green" title="E37">A51</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E5:49_5_1"/><span class="lab"><font color="Green" title="E40">A54</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_5"><span class="lab"><font color="Green" title="E37">A51</font></span></a>, <a class="txt" href="#E3:49_5_1"><span class="lab"><font color="Green" title="E38">A52</font></span></a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E3:49_5_1"><span class="lab"><font color="Green" title="E38">A52</font></span></a>, <a class="ref" href="card_3.html#D1" target="_self" title="CARD_3:def.1">Def1</a></span>;<br/><a NAME="E7:49_5_1"/><span class="lab"><font color="Green" title="E41">A55</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T20" target="_self" title="CARD_3:th.20">Th31</a></span>;<br/><a NAME="E8:49_5_1"/><span class="lab"><font color="Green" title="E42">A56</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T67" title="CARD_1:th.67">CARD_1:67</a></span>;<br/><a NAME="E9:49_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_5_1"><span class="lab"><font color="Green" title="E38">A52</font></span></a>, <a class="ref" href="card_3.html#T29" target="_self" title="CARD_3:th.29">Th40</a></span>;<br/><a NAME="E10:49_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A57</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_5_1"><span class="lab"><font color="Green" title="E41">A55</font></span></a>, <a class="txt" href="#E8:49_5_1"><span class="lab"><font color="Green" title="E42">A56</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E11:49_5_1"/><span class="lab"><font color="Green" title="E44">A58</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:49_5_1"><span class="lab"><font color="Green" title="E38">A52</font></span></a></span>;<br/><a NAME="E12:49_5_1"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E13:49_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_5_1"><span class="lab"><font color="Green" title="E39">A53</font></span></a>, <a class="txt" href="#E5:49_5_1"><span class="lab"><font color="Green" title="E40">A54</font></span></a>, <a class="txt" href="#E10:49_5_1"><span class="lab"><font color="Green" title="E43">A57</font></span></a>, <a class="txt" href="#E11:49_5_1"><span class="lab"><font color="Green" title="E44">A58</font></span></a>, <a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:49_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A59</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T26" target="_self" title="CARD_3:th.26">Th37</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">t</font> =  the    <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/><span class="kw">consider </span><font color="Maroon" title="c10">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E12:49_5"/><span class="lab"><font color="Green" title="E39">A60</font></span>: 
 the    <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">and </span><br/><a NAME="E13:49_5"/>
 <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>
 <span class="kw">and </span><br/><a NAME="E14:49_5"/><span class="lab"><font color="Green" title="E40">A61</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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_5"><span class="lab"><font color="Green" title="E38">A59</font></span></a>, <a class="ref" href="card_3.html#D5" target="_self" title="CARD_3:def.5">Def5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E1:49_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:49_5_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_5"><span class="lab"><font color="Green" title="E37">A51</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:49_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E16:49_5"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E8:49_5"><span class="lab"><font color="Green" title="E37">A51</font></span></a>, <a class="ref" href="card_3.html#T27" target="_self" title="CARD_3:th.27">Th38</a></span>;<br/><a NAME="E17:49_5"/><span class="kw">then </span>
 the    <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> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_5"><span class="lab"><font color="Green" title="E38">A59</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E19:49_5"/><span class="lab"><font color="Green" title="E41">A62</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E20:49_5"/><span class="lab"><font color="Green" title="E42">A63</font></span>: 
 the    <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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_5"><span class="lab"><font color="Green" title="E36">A50</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E22:49_5"/><span class="lab"><font color="Green" title="E43">A64</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E23:49_5"/><span class="lab"><font color="Green" title="E44">A65</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_5"><span class="lab"><font color="Green" title="E35">A49</font></span></a>, <a class="txt" href="#E19:49_5"><span class="lab"><font color="Green" title="E41">A62</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E25:49_5"/><span class="lab"><font color="Green" title="E45">A66</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E26:49_5"/><span class="lab"><font color="Green" title="E46">A67</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:49_5"><span class="lab"><font color="Green" title="E44">A65</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E27:49_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:49_5"><span class="lab"><font color="Green" title="E39">A60</font></span></a>, <a class="txt" href="#E19:49_5"><span class="lab"><font color="Green" title="E41">A62</font></span></a>, <a class="txt" href="#E20:49_5"><span class="lab"><font color="Green" title="E42">A63</font></span></a>, <a class="txt" href="#E22:49_5"><span class="lab"><font color="Green" title="E43">A64</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><a NAME="E28:49_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A68</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:49_5"><span class="lab"><font color="Green" title="E46">A67</font></span></a>, <a class="ref" href="card_3.html#D6" target="_self" title="CARD_3:def.6">Def6</a></span>;<br/><a NAME="E29:49_5"/><span class="lab"><font color="Green" title="E48">A69</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:49_5"><span class="lab"><font color="Green" title="E45">A66</font></span></a>, <a class="ref" href="card_3.html#D3" target="_self" title="CARD_3:def.3">Def3</a></span>;<br/><a NAME="E30:49_5"/><span class="lab"><font color="Green" title="E49">A70</font></span>: 
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:49_5"><span class="lab"><font color="Green" title="E47">A68</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E31:49_5"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_5"><span class="lab"><font color="Green" title="E37">A51</font></span></a>, <a class="txt" href="#E14:49_5"><span class="lab"><font color="Green" title="E40">A61</font></span></a>, <a class="txt" href="#E29:49_5"><span class="lab"><font color="Green" title="E48">A69</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E32:49_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_5"><span class="lab"><font color="Green" title="E37">A51</font></span></a>, <a class="txt" href="#E29:49_5"><span class="lab"><font color="Green" title="E48">A69</font></span></a>, <a class="txt" href="#E30:49_5"><span class="lab"><font color="Green" title="E49">A70</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E18:49"/>
 <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E16:49"><span class="lab"><font color="Green" title="E34">A48</font></span></a>, <a class="ref" href="card_1.html#T3" title="CARD_1:th.3">CARD_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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