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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>  st <font color="Maroon" title="c1">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>  holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c2">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>  implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c1">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c2">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font>;<br/>
<div><span class="lab"><font color="Green" title="E36">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:50_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
 <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:50_1_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">and </span><br/><a NAME="E3:50_1_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T41" title="ZFMISC_1:th.41">ZFMISC_1:41</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">C</font> = <font color="Maroon" title="c4">C</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">b</font> =  <a href="ens_1.html#K13" title="ENS_1:func.13">@</a> <font color="Maroon" title="c5">C</font>;<br/><a NAME="E5:50_1_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K13" title="ENS_1:func.13">@</a> <font color="Maroon" title="c5">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="cat_1.html#D15" title="CAT_1:def.15">CAT_1:def 15</a></span>;<br/><a NAME="E6:50_1_1"/><span class="kw">then </span>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K13" title="ENS_1:func.13">@</a> <font color="Maroon" title="c5">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="ens_1.html#E51"><span class="lab"><font color="Green" title="E30">Lm6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:50_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E3:50_1_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:50_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:50_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:50_2"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E4:50_2"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T41" title="ZFMISC_1:th.41">ZFMISC_1:41</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">fy</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c4">y</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">fx</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c3">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></span>;<br/><a NAME="E7:50_2"/>
<font color="Maroon" title="c6">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E8:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c6">fx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">fy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E4:50_2"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E9:50_2"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c6">fx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T15" target="_self" title="ENS_1:th.15">Th15</a></span>;<br/><a NAME="E10:50_2"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c5">fy</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T15" target="_self" title="ENS_1:th.15">Th15</a></span>;<br/><a NAME="E11:50_2"/>
 <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T17" target="_self" title="ENS_1:th.17">Th17</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">m1</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c6">fx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>, <font color="Maroon" title="c8">m2</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c5">fy</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E10:50_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a></span>;<br/><a NAME="E13:50_2"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c8">m2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="ens_1.html#T27" target="_self" title="ENS_1:th.27">Th27</a></span>;<br/><a NAME="E14:50_2"/>
<font color="Maroon" title="c7">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="ens_1.html#T27" target="_self" title="ENS_1:th.27">Th27</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> =  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c7">m1</font>, <font color="Maroon" title="c10">g</font> =  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c8">m2</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:50_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">h</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font><span class="kw"> such that </span><br/><a NAME="E17:50_2"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
for <font color="Olive" title="b1">h9</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="cat_1.html#D15" title="CAT_1:def.15">CAT_1:def 15</a></span>;<br/><font color="Maroon" title="c9">f</font> = 
<font color="Maroon" title="c11">h</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:50_2"><span class="lab"><font color="Green" title="E43">A13</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:50_2"><span class="lab"><font color="Green" title="E43">A13</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E19:50_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:50"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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