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

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">Y</font> implies  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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="E3:25"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
( <font color="Maroon" title="c3">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" onmouseover="rs('card_1/T5')" onmouseout="rh()">CARD_1:5</a>, <a class="ref" href="wellord2.html#D4" onmouseover="rs('wellord2/D4')" onmouseout="rh()">WELLORD2:def 4</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of $1 =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of $1;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of $1 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>($1)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of $1 <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>($1)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> ($1,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> ($1,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">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="E5:25"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c1">X</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c4">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>3</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" onmouseover="rs('funct_1/S3')" onmouseout="rh()">FUNCT_1:sch 3</a>();<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> implies <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font> ) &amp; ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font> implies <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> implies <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font> ) &amp; ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:25_1_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">X9</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:25_1_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X9</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">X9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E3:25_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E11">A5</font></span>: <font color="Maroon" title="c5">y</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c7">X9</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c7">X9</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E3:25_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E5:25_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>

;<br/><a NAME="E7:25_1_1"/>
 not <font color="Maroon" title="c7">X9</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_1_1_2"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c7">X9</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:25_1_1_2_1"/>
( <font color="Maroon" title="c7">X9</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c7">X9</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_1_1_2_1_1"/>
<font color="Maroon" title="c7">X9</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:25_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_1_1_2_1_2"/>
 not <font color="Maroon" title="c7">X9</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:25_1_1_2_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">X9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_2"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E8:25_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">X9</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>  &amp; <font color="Maroon" title="c7">X9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="zfmisc_1.html#T139" onmouseover="rs('zfmisc_1/T139')" onmouseout="rh()">ZFMISC_1:139</a></span>;<br/><a NAME="E9:25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A7</font></span>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">X9</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">X9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><a NAME="E10:25_1_1"/><span class="kw">then </span>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">X9</font> &amp;  not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E11:25_1_1"/><span class="kw">then </span>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">X9</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E12:25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E9:25_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><a NAME="E13:25_1_1"/>
( <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E9:25_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E14:25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T32" onmouseover="rs('zfmisc_1/T32')" onmouseout="rh()">ZFMISC_1:32</a></span>;<br/><a NAME="E15:25_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X9</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X9</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="card_2.html#T57" onmouseover="rs('card_2/T57')" onmouseout="rh()">CARD_2:57</a></span>;<br/><span class="kw">hence </span><a NAME="E16:25_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E14:25_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">assume </span><a NAME="E2:25_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">Y9</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E4:25_1"/><span class="lab"><font color="Green" title="E9">A10</font></span>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y9</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">Y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">y1</font>, <font color="Maroon" title="c8">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:25_1"/><span class="lab"><font color="Green" title="E10">A11</font></span>: 
( <font color="Maroon" title="c7">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y2</font> &amp; <font color="Maroon" title="c6">Y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c8">y2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="ref" href="card_2.html#T60" onmouseover="rs('card_2/T60')" onmouseout="rh()">CARD_2:60</a></span>;<br/><a NAME="E7:25_1"/>
( <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y9</font> &amp; <font color="Maroon" title="c8">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E8:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A12</font></span>: 
( <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c8">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:25_1"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E12:25_1"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25_1"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E13:25_1"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E10:25_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E12:25_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">X9</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E10:25_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E12:25_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" onmouseover="rs('zfmisc_1/T32')" onmouseout="rh()">ZFMISC_1:32</a></span>;<br/><a NAME="E15:25_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c11">X9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="card_2.html#T57" onmouseover="rs('card_2/T57')" onmouseout="rh()">CARD_2:57</a></span>;<br/><a NAME="E16:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A16</font></span>: 
<font color="Maroon" title="c11">X9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c1">X</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:25_1_2"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> or <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:25_1_2_1"/><span class="lab"><font color="Green" title="E16">A17</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:25_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">X9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X9</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T17" onmouseover="rs('zfmisc_1/T17')" onmouseout="rh()">ZFMISC_1:17</a></span>;<br/><a NAME="E3:25_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A18</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">X9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">X9</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c11">X9</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c11">X9</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E16:25_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E10:25_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E12:25_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E1:25_1_2_1"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="txt" href="#E3:25_1_2_1"><span class="lab"><font color="Green" title="E17">A18</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:25_1_2_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E16:25_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:25_1_2_2"/><span class="lab"><font color="Green" title="E16">A19</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:25_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">X9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X9</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T17" onmouseover="rs('zfmisc_1/T17')" onmouseout="rh()">ZFMISC_1:17</a></span>;<br/><a NAME="E3:25_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A20</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">X9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">X9</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c11">X9</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c11">X9</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E16:25_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="txt" href="#E10:25_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E12:25_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E1:25_1_2_2"><span class="lab"><font color="Green" title="E16">A19</font></span></a>, <a class="txt" href="#E3:25_1_2_2"><span class="lab"><font color="Green" title="E17">A20</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:25_1_2_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E16:25_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E7:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A21</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:25_2"/><span class="lab"><font color="Green" title="E10">A22</font></span>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">X1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:25_2"/><span class="lab"><font color="Green" title="E11">A23</font></span>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">X2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:25_2"/><span class="lab"><font color="Green" title="E12">A24</font></span>: 
( <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:25_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E13">A25</font></span>: <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c7">X1</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c7">X1</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:25_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>

;<br/><span class="lab"><font color="Green" title="E14">A26</font></span>: <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c8">X2</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>( <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c8">X2</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>))</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:25_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E12">A24</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>

;<br/><a NAME="E8:25_2"/>
 not <font color="Maroon" title="c7">X1</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_2_3"/><span class="lab"><font color="Green" title="E15">A27</font></span>: 
<font color="Maroon" title="c7">X1</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:25_2_3_1"/>
( <font color="Maroon" title="c7">X1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c7">X1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_2_3_1_1"/>
<font color="Maroon" title="c7">X1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:25_2_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_2_3_1_2"/>
 not <font color="Maroon" title="c7">X1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:25_2_3_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2_3"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_2_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E9:25_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">X1</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>  &amp; <font color="Maroon" title="c7">X1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="zfmisc_1.html#T139" onmouseover="rs('zfmisc_1/T139')" onmouseout="rh()">ZFMISC_1:139</a></span>;<br/><a NAME="E10:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A28</font></span>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">X1</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">X1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><a NAME="E11:25_2"/><span class="kw">then </span>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">X1</font> &amp;  not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E12:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A29</font></span>: 
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">X1</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E13:25_2"/><span class="lab"><font color="Green" title="E17">A30</font></span>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25_2"><span class="lab"><font color="Green" title="E15">A28</font></span></a></span>;<br/><a NAME="E14:25_2"/>
 not <font color="Maroon" title="c8">X2</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_2_4"/><span class="lab"><font color="Green" title="E18">A31</font></span>: 
<font color="Maroon" title="c8">X2</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_4_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:25_2_4_1"/>
( <font color="Maroon" title="c8">X2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c8">X2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_2_4_1_1"/>
<font color="Maroon" title="c8">X2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:25_2_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E12">A24</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_2_4_1_2"/>
 not <font color="Maroon" title="c8">X2</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:25_2_4_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2_4"><span class="lab"><font color="Green" title="E18">A31</font></span></a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_2_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E12">A24</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E15:25_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">X2</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>  &amp; <font color="Maroon" title="c8">X2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="zfmisc_1.html#T139" onmouseover="rs('zfmisc_1/T139')" onmouseout="rh()">ZFMISC_1:139</a></span>;<br/><a NAME="E16:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A32</font></span>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">X2</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">X2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/><a NAME="E17:25_2"/><span class="kw">then </span>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">X2</font> &amp;  not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E18:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A33</font></span>: 
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">X2</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E19:25_2"/><span class="lab"><font color="Green" title="E20">A34</font></span>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:25_2"><span class="lab"><font color="Green" title="E18">A32</font></span></a></span>;<br/><a NAME="E20:25_2"/><span class="lab"><font color="Green" title="E21">A35</font></span>: 
( <font color="Maroon" title="c7">X1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Maroon" title="c8">X2</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E12">A24</font></span></a></span>;<br/><a NAME="E21:25_2"/>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X1</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25_2"><span class="lab"><font color="Green" title="E16">A29</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" onmouseover="rs('zfmisc_1/T32')" onmouseout="rh()">ZFMISC_1:32</a>, <a class="ref" href="card_2.html#T57" onmouseover="rs('card_2/T57')" onmouseout="rh()">CARD_2:57</a></span>;<br/><a NAME="E22:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A36</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E20:25_2"><span class="lab"><font color="Green" title="E21">A35</font></span></a>, <a class="ref" href="card_2.html#T102" onmouseover="rs('card_2/T102')" onmouseout="rh()">CARD_2:102</a></span>;<br/><a NAME="E23:25_2"/>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:25_2"><span class="lab"><font color="Green" title="E19">A33</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" onmouseover="rs('zfmisc_1/T32')" onmouseout="rh()">ZFMISC_1:32</a>, <a class="ref" href="card_2.html#T57" onmouseover="rs('card_2/T57')" onmouseout="rh()">CARD_2:57</a></span>;<br/><a NAME="E24:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A37</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E12">A24</font></span></a>, <a class="txt" href="#E20:25_2"><span class="lab"><font color="Green" title="E21">A35</font></span></a>, <a class="ref" href="card_2.html#T102" onmouseover="rs('card_2/T102')" onmouseout="rh()">CARD_2:102</a></span>;<br/><a NAME="E25:25_2"/><span class="lab"><font color="Green" title="E24">A38</font></span>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Y1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E27:25_2"/><span class="lab"><font color="Green" title="E25">A39</font></span>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y1</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c9">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E9">A21</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">Y2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E29:25_2"/><span class="lab"><font color="Green" title="E26">A40</font></span>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Y2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c10">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="txt" href="#E25:25_2"><span class="lab"><font color="Green" title="E24">A38</font></span></a></span>;<br/><a NAME="E30:25_2"/><span class="lab"><font color="Green" title="E27">A41</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_2_5"/>
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:25_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_2"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:25_2_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:25_2"><span class="lab"><font color="Green" title="E25">A39</font></span></a>, <a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E31:25_2"/><span class="lab"><font color="Green" title="E28">A42</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:25_2_6"/>
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:25_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_2"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:25_2_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:25_2"><span class="lab"><font color="Green" title="E26">A40</font></span></a>, <a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E32:25_2"/><span class="lab"><font color="Green" title="E29">A43</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E6:25_2"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E7:25_2"><span class="lab"><font color="Green" title="E14">A26</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_7"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:25_2_7"/><span class="kw">then </span>
( <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>) or <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T6" onmouseover="rs('zfmisc_1/T6')" onmouseout="rh()">ZFMISC_1:6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_7_1"><span class="kw">suppose </span></a><a NAME="E1:25_2_7_1"/><span class="lab"><font color="Green" title="E30">A44</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:25_2_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A45</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E13:25_2"><span class="lab"><font color="Green" title="E17">A30</font></span></a>, <a class="txt" href="#E19:25_2"><span class="lab"><font color="Green" title="E20">A34</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><a NAME="E3:25_2_7_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:25_2"><span class="lab"><font color="Green" title="E27">A41</font></span></a>, <a class="ref" href="zfmisc_1.html#T17" onmouseover="rs('zfmisc_1/T17')" onmouseout="rh()">ZFMISC_1:17</a></span>;<br/><a NAME="E4:25_2_7_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:25_2"><span class="lab"><font color="Green" title="E28">A42</font></span></a>, <a class="txt" href="#E32:25_2"><span class="lab"><font color="Green" title="E29">A43</font></span></a>, <a class="txt" href="#E1:25_2_7_1"><span class="lab"><font color="Green" title="E30">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#T17" onmouseover="rs('zfmisc_1/T17')" onmouseout="rh()">ZFMISC_1:17</a></span>;<br/><a NAME="E5:25_2_7_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E10:25_2"><span class="lab"><font color="Green" title="E15">A28</font></span></a>, <a class="txt" href="#E16:25_2"><span class="lab"><font color="Green" title="E18">A32</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a>, <a class="ref" href="zfmisc_1.html#T3" onmouseover="rs('zfmisc_1/T3')" onmouseout="rh()">ZFMISC_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:25_2_7_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E12">A24</font></span></a>, <a class="txt" href="#E22:25_2"><span class="lab"><font color="Green" title="E22">A36</font></span></a>, <a class="txt" href="#E24:25_2"><span class="lab"><font color="Green" title="E23">A37</font></span></a>, <a class="txt" href="#E2:25_2_7_1"><span class="lab"><font color="Green" title="E31">A45</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_7_2"><span class="kw">suppose </span></a><a NAME="E1:25_2_7_2"/><span class="lab"><font color="Green" title="E30">A46</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:25_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A47</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">X2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E10:25_2"><span class="lab"><font color="Green" title="E15">A28</font></span></a>, <a class="txt" href="#E16:25_2"><span class="lab"><font color="Green" title="E18">A32</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><a NAME="E3:25_2_7_2"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:25_2"><span class="lab"><font color="Green" title="E27">A41</font></span></a>, <a class="ref" href="zfmisc_1.html#T17" onmouseover="rs('zfmisc_1/T17')" onmouseout="rh()">ZFMISC_1:17</a></span>;<br/><a NAME="E4:25_2_7_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:25_2"><span class="lab"><font color="Green" title="E28">A42</font></span></a>, <a class="txt" href="#E32:25_2"><span class="lab"><font color="Green" title="E29">A43</font></span></a>, <a class="txt" href="#E1:25_2_7_2"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="ref" href="zfmisc_1.html#T17" onmouseover="rs('zfmisc_1/T17')" onmouseout="rh()">ZFMISC_1:17</a></span>;<br/><a NAME="E5:25_2_7_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">X1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">X2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E10:25_2"><span class="lab"><font color="Green" title="E15">A28</font></span></a>, <a class="txt" href="#E16:25_2"><span class="lab"><font color="Green" title="E18">A32</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a>, <a class="ref" href="zfmisc_1.html#T3" onmouseover="rs('zfmisc_1/T3')" onmouseout="rh()">ZFMISC_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:25_2_7_2"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E12">A24</font></span></a>, <a class="txt" href="#E22:25_2"><span class="lab"><font color="Green" title="E22">A36</font></span></a>, <a class="txt" href="#E24:25_2"><span class="lab"><font color="Green" title="E23">A37</font></span></a>, <a class="txt" href="#E2:25_2_7_2"><span class="lab"><font color="Green" title="E31">A47</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:25"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a>, <a class="ref" href="card_1.html#T70" onmouseover="rs('card_1/T70')" onmouseout="rh()">CARD_1:70</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
