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

<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> is  <a href="card_fil.html#V7" title="CARD_FIL:attr.7">measurable</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="card_fil.html#V5" title="CARD_FIL:attr.5">strong_limit</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:70"/>
<font color="Maroon" title="c1">M</font> is  <a href="card_fil.html#V7" title="CARD_FIL:attr.7">measurable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> is  <a href="card_fil.html#V5" title="CARD_FIL:attr.5">strong_limit</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">F</font> being    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E3:70"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="card_fil.html#NR3" title="CARD_FIL:NR.3">is_complete_with</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">and </span><br/><a NAME="E4:70"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
(  not <font color="Maroon" title="c2">F</font> is  <a href="card_fil.html#V2" title="CARD_FIL:attr.2">principal</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fil.html#D16" target="_self" title="CARD_FIL:def.16">Def16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:70"/>
 not <font color="Maroon" title="c1">M</font> is  <a href="card_fil.html#V5" title="CARD_FIL:attr.5">strong_limit</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">N</font> being   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a><span class="kw"> such that </span><br/><a NAME="E7:70"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<font color="Maroon" title="c3">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">and </span><br/><a NAME="E8:70"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
 not  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<font color="Maroon" title="c3">N</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fil.html#D14" target="_self" title="CARD_FIL:def.14">Def14</a></span>;<br/>
<a NAME="E9:70"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<font color="Maroon" title="c3">N</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="card_1.html#T4" title="CARD_1:th.4">CARD_1:4</a></span>;<br/>
<a NAME="E10:70"/><span class="kw">then </span>
<font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c3">N</font>,2)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#D3" title="CARD_2:def.3">CARD_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:70"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c3">N</font>,2)
 <span class="kw">and </span><br/><a NAME="E13:70"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fil.html#T36" target="_self" title="CARD_FIL:th.36">Th36</a></span>;<br/>
<a NAME="E14:70"/>
 not <font color="Maroon" title="c3">N</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:70_1"/>
<font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> (2,<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">N</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<a NAME="E2:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A8</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c3">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T31" title="CARD_2:th.31">CARD_2:31</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:70_1"/>
<font color="Maroon" title="c3">N</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E4:70_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E54">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">N1</font> = <font color="Maroon" title="c3">N</font> as   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a> ;<br/>
<a NAME="E16:70"/>
<font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">M</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:70"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">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="E18:70"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E19:70"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">and </span><br/><a NAME="E20:70"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><font color="Maroon" title="c6">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 ) </span> } </span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>;<br/>
<a NAME="E21:70"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
 for <font color="Olive" title="b1">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font>  ex <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">A</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">A</font>,<font color="Olive" title="b1">i</font>]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">Y1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Inv1</font> = <font color="Maroon" title="c6">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="relat_1.html#T132" title="RELAT_1:th.132">RELAT_1:132</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Y0</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Inv0</font> = <font color="Maroon" title="c6">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="relat_1.html#T132" title="RELAT_1:th.132">RELAT_1:132</a></span>;<br/>
<a NAME="E3:70_2"/><span class="lab"><font color="Green" title="E58">A13</font></span>: 
 for <font color="Olive" title="b1">g1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>  holds <br/>(  not <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> or <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">h</font> where <font color="Olive" title="b2">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  or <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">h</font> where <font color="Olive" title="b2">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">g1</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> or <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  or <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_2_1"/><span class="lab"><font color="Green" title="E59">A14</font></span>: 
<font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  or <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_2_1_1"/>
( <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  or  not <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_2_1_1_1"/>
<font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  or <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_2_1_1_1"/>
( <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  or <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_2_1_1_2"/>
 not <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  or <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )</span><br/><div class="add"><a NAME="E2:70_2_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c12">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_2_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a></span>;<br/><a NAME="E3:70_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:70_2_1_1_2"/>
( <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  or <font color="Maroon" title="c12">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_2_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:70_2"/>
<font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:70_2_2"/>
<font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  or <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_2_2_1"/><span class="lab"><font color="Green" title="E59">A15</font></span>: 
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span> </span><br/>

<a NAME="E2:70_2_2_1"/>
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_2_2_1"><span class="lab"><font color="Green" title="E59">A15</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">g1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:70_2_2_1"/><span class="lab"><font color="Green" title="E60">A16</font></span>: 
<font color="Maroon" title="c13">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">X</font>
 <span class="kw">and </span><br/><a NAME="E5:70_2_2_1"/><span class="lab"><font color="Green" title="E61">A17</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">g2</font> = <font color="Maroon" title="c13">g1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_2_2_1"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E7:70_2_2_1"/>
 not <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_2_2_1"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E8:70_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_2"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E1:70_2_2_1"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="#E4:70_2_2_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:70_2_2_1"/>
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_2_2_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c12">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span>  or <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:70_2_2"/>
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span><span class="kw"> such that </span><br/><a NAME="E4:70_2_2"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
( <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> &amp; <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )
 <span class="kw">and </span><br/><a NAME="E5:70_2_2"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>
<a NAME="E6:70_2_2"/>
 not <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:70_2_2_2"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:70_2_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">h1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> st <br/>( <font color="Olive" title="b1">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">h</font> &amp; <font color="Olive" title="b1">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:70_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_2_2"><span class="lab"><font color="Green" title="E60">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:70_2_2"/>
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_2_2"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="lab"><font color="Green" title="E59">A20</font></span>: <font color="Maroon" title="c9">Inv1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c11">Inv0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:70"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="funct_1.html#T69" title="FUNCT_1:th.69">FUNCT_1:69</a></span>
<br/>.= 
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c11">Inv0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="relat_1.html#T134" title="RELAT_1:th.134">RELAT_1:134</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_2_4"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_2_4"/>
( <font color="Maroon" title="c11">Inv0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> or <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c11">Inv0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="card_fil.html#D7" target="_self" title="CARD_FIL:def.7">Def7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_2_4_1"><span class="kw">suppose </span></a><a NAME="E1:70_2_4_1"/><span class="lab"><font color="Green" title="E60">A21</font></span>: 
<font color="Maroon" title="c11">Inv0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">A</font>,<font color="Olive" title="b1">i</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">Z</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c12">Z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c12">Z</font>]</span><br/><span class="kw">thus </span><a NAME="E3:70_2_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c12">Z</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_2_4_1"><span class="lab"><font color="Green" title="E60">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_2_4_2"><span class="kw">suppose </span></a><a NAME="E1:70_2_4_2"/><span class="lab"><font color="Green" title="E60">A22</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c11">Inv0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">A</font>,<font color="Olive" title="b1">i</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">O</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c12">O</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c12">O</font>]</span><br/><span class="kw">thus </span><a NAME="E3:70_2_4_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c12">O</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_2"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="txt" href="#E1:70_2_4_2"><span class="lab"><font color="Green" title="E60">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span><span class="kw"> such that </span><br/><a NAME="E23:70"/><span class="lab"><font color="Green" title="E58">A23</font></span>: 
 for <font color="Olive" title="b1">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">A</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E21:70"><span class="lab"><font color="Green" title="E57">A12</font></span></a>);<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c6">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 ) </span> } </span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c6">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E20:70"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="funct_2.html#T1" title="FUNCT_2:th.1">FUNCT_2:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">MEET</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> ;<br/>
<a NAME="E25:70"/><span class="lab"><font color="Green" title="E59">A24</font></span>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> </span>)</span> &amp; <font color="Maroon" title="c6">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> </span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="funct_1.html#T52" title="FUNCT_1:th.52">FUNCT_1:52</a>, <a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</a></span>;<br/>
<a NAME="E26:70"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">N1</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="trees_2.html#S2" title="TREES_2:sch.2">TREES_2:sch 2</a>();<br/></span>
<a NAME="E27:70"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<a NAME="E28:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A25</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">B</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font>;<br/>
<a NAME="E29:70"/><span class="lab"><font color="Green" title="E61">A26</font></span>: 
<span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  or <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_3"/>
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E2:70_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">A</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> st <br/>( <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:70_3"/>
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:70"><span class="lab"><font color="Green" title="E58">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:70"/>
<font color="Maroon">H<sub>1</sub></font>( the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> 
 
;<br/>
<a NAME="E31:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A27</font></span>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E28:70"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="txt" href="#E29:70"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="ref" href="card_fil.html#D3" target="_self" title="CARD_FIL:def.3">Def3</a></span>;<br/>
<a NAME="E32:70"/><span class="lab"><font color="Green" title="E63">A28</font></span>: 
 not <font color="Maroon" title="c4">Y</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="#E13:70"><span class="lab"><font color="Green" title="E53">A7</font></span></a></span>;<br/>
<a NAME="E33:70"/><span class="lab"><font color="Green" title="E64">A29</font></span>: 
 for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">A</font>) where <font color="Olive" title="b2">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  holds <br/><font color="Maroon" title="c6">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" title="c7">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  implies <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_4"/><span class="lab"><font color="Green" title="E65">A30</font></span>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">X1</font> = <font color="Maroon" title="c11">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:70"><span class="lab"><font color="Green" title="E62">A27</font></span></a></span>;<br/>
<a NAME="E3:70_4"/>
<font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:70"><span class="lab"><font color="Green" title="E63">A28</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">h1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:70_4"/><span class="lab"><font color="Green" title="E66">A31</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h1</font>
 <span class="kw">and </span><br/><a NAME="E6:70_4"/><span class="lab"><font color="Green" title="E67">A32</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E7:70_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">h1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E8:70_4"/><span class="lab"><font color="Green" title="E68">A33</font></span>: 
 for <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> holds <br/><font color="Maroon" title="c13">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> implies <font color="Maroon" title="c13">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">Z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_4_1"/>
<font color="Maroon" title="c14">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">Z</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">Z1</font> = <font color="Maroon" title="c14">Z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> ;<br/>
<a NAME="E3:70_4_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">Z1</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> 
 
;<br/>
<a NAME="E4:70_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c15">Z1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_4"><span class="lab"><font color="Green" title="E65">A30</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E5:70_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">Z1</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E6:70_4_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">N1</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> st <br/>( <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">Z1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E7:70_4_1"/>
<font color="Maroon" title="c13">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_4"><span class="lab"><font color="Green" title="E66">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:70_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:70_4"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_4"><span class="lab"><font color="Green" title="E66">A31</font></span></a>, <a class="txt" href="#E6:70_4"><span class="lab"><font color="Green" title="E67">A32</font></span></a>, <a class="txt" href="#E8:70_4"><span class="lab"><font color="Green" title="E68">A33</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E34:70"/><span class="lab"><font color="Green" title="E65">A34</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">g</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> </span>)</span> or <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">g</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_5"/>
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">g</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:70_5"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">A</font>) where <font color="Olive" title="b2">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  &amp; <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E3:70_5"/><span class="kw">then </span>
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:70"><span class="lab"><font color="Green" title="E64">A29</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:70_5"/>
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">g</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E35:70"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E31:70"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E36:70"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E37:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A35</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span> </span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">g</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="#E34:70"><span class="lab"><font color="Green" title="E65">A34</font></span></a>, <a class="txt" href="#E25:70"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="ref" href="card_1.html#T10" title="CARD_1:th.10">CARD_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">MEET</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>) where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">N1</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font> </span> } </span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:70"><span class="lab"><font color="Green" title="E62">A27</font></span></a></span>;<br/>
<a NAME="E39:70"/>
<font color="Maroon" title="c2">F</font> <a href="card_fil.html#NR3" title="CARD_FIL:NR.3">is_complete_with</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<a NAME="E40:70"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> is  <a href="card_fil.html#V1" title="CARD_FIL:attr.1">uniform</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="card_fil.html#T23" target="_self" title="CARD_FIL:th.23">Th23</a></span>;<br/>
<a NAME="E41:70"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c11">MEET</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="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:70"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="ref" href="card_fil.html#D5" target="_self" title="CARD_FIL:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E42:70"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:70"><span class="lab"><font color="Green" title="E66">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
