<?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">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</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="Olive" title="b1">f</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>  &amp;  ex <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ex <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b3">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">x2</font> ) holds <br/> <a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Olive" title="b1">f</font> is  <a href="ens_1.html#V1" title="ENS_1:attr.1">surjective</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</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="c2">f</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>  &amp;  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="c1">V</font> ex <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">x2</font> ) implies  <a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font> is  <a href="ens_1.html#V1" title="ENS_1:attr.1">surjective</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:53"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> or  not <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> or  not <font color="Olive" title="b2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">x2</font> ) or  <a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font> is  <a href="ens_1.html#V1" title="ENS_1:attr.1">surjective</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c3">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E3:53"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E4:53"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E5:53"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font> is  <a href="ens_1.html#V1" title="ENS_1:attr.1">surjective</a> </span><br/>

<a NAME="E6:53"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">A</font> =  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">g1</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c4">x1</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#K4" title="ENS_1:func.4">cod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E36">A2</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">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:53_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:53_1"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</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">set </span><font color="Maroon" title="c9">h</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">g2</font> = <font color="Maroon" title="c7">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</font></span>)</span>;<br/>
<a NAME="E4:53_1"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</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="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E5:53_1"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></span>;<br/>
<a NAME="E6:53_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g1</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="Maroon" title="c4">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></span>;<br/>
<a NAME="E7:53_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</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="c4">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_1"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></span>;<br/>
<a NAME="E8:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A10</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="c7">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></span>;<br/>
<a NAME="E9:53_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></span>;<br/>
<a NAME="E10:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A11</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="c7">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:53_1"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E11:53_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<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="c7">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</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="#E4:53_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">g2</font> = <font color="Maroon" title="c7">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</font></span>)</span> 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#K4" title="ENS_1:func.4">cod</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E10:53_1"><span class="lab"><font color="Green" title="E45">A11</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#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<a NAME="E14:53_1"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></span>;<br/>
<a NAME="E15:53_1"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
<font color="Maroon" title="c8">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="c8">x</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>;<br/>
<a NAME="E16:53_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E17:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A14</font></span>: 
<font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:53_1"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>;<br/>
<a NAME="E18:53_1"/><span class="lab"><font color="Green" title="E49">A15</font></span>: 
<font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">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#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c7">g1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>, <font color="Maroon" title="c13">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#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c11">g2</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="#E3:53"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="ens_1.html#T5" target="_self" title="ENS_1:th.5">Th5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">f1</font> =  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">f2</font> =  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">h1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">h2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">f1f</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">f2f</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<a NAME="E20:53_1"/><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<font color="Maroon" title="c12">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><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#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">m1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T8" target="_self" title="ENS_1:th.8">Th8</a></span>;<br/>
<a NAME="E21:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A17</font></span>: 
<font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E22:53_1"/><span class="lab"><font color="Green" title="E52">A18</font></span>: 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c12">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:53_1"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="ens_1.html#E10"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<a NAME="E23:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A19</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<a NAME="E24:53_1"/><span class="lab"><font color="Green" title="E54">A20</font></span>: 
<font color="Maroon" title="c13">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><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#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c13">m2</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c13">m2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T8" target="_self" title="ENS_1:th.8">Th8</a></span>;<br/>
<a NAME="E25:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A21</font></span>: 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c13">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ens_1.html#E10"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<a NAME="E26:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A22</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<a NAME="E27:53_1"/>
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:53_1"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="txt" href="ens_1.html#E10"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<a NAME="E28:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A23</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></span>;<br/>
<a NAME="E29:53_1"/>
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:53_1"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="txt" href="ens_1.html#E10"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<a NAME="E30:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A24</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a></span>;<br/>
<a NAME="E31:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A25</font></span>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:53_1"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="ens_1.html#T28" target="_self" title="ENS_1:th.28">Th28</a></span>;<br/>
<a NAME="E32:53_1"/><span class="lab"><font color="Green" title="E60">A26</font></span>: 
<font color="Maroon" title="c11">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:53_1"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">thus </span><a NAME="E1:53_1_2"/><span class="lab"><font color="Green" title="E61">A27</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:53_1"><span class="lab"><font color="Green" title="E53">A19</font></span></a>, <a class="txt" href="#E26:53_1"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="txt" href="ens_1.html#E17"><span class="lab"><font color="Green" title="E10">Lm3</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"> 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>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c20">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c21">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>;<br/><span class="kw">assume </span><a NAME="E2:53_1_2"/><span class="lab"><font color="Green" title="E62">A28</font></span>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font></span><br/><a NAME="E3:53_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="ens_1.html#E17"><span class="lab"><font color="Green" title="E10">Lm3</font></span></a></span>;<br/><a NAME="E4:53_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:53_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A29</font></span>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</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="c8">x</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="#E3:53_1"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:53_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:53_1"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="txt" href="#E32:53_1"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="txt" href="#E1:53_1_2"><span class="lab"><font color="Green" title="E61">A27</font></span></a>, <a class="txt" href="#E2:53_1_2"><span class="lab"><font color="Green" title="E62">A28</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/><span class="kw">hence </span><a NAME="E7:53_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E5:53_1_2"><span class="lab"><font color="Green" title="E63">A29</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</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>
<a NAME="E34:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A30</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<a NAME="E35:53_1"/>
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:53_1"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<a NAME="E36:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A31</font></span>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><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#K3" title="ENS_1:func.3">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:53_1"><span class="lab"><font color="Green" title="E53">A19</font></span></a>, <a class="ref" href="ens_1.html#T8" target="_self" title="ENS_1:th.8">Th8</a></span>;<br/>
<a NAME="E37:53_1"/>
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:53_1"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<a NAME="E38:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A32</font></span>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><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#K3" title="ENS_1:func.3">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:53_1"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="ref" href="ens_1.html#T8" target="_self" title="ENS_1:th.8">Th8</a></span>;<br/>
<a NAME="E39:53_1"/>
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c12">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:53_1"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="ens_1.html#E10"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<a NAME="E40:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A33</font></span>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:53_1"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="txt" href="#E36:53_1"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="txt" href="#E38:53_1"><span class="lab"><font color="Green" title="E63">A32</font></span></a>, <a class="txt" href="#E34:53_1"><span class="lab"><font color="Green" title="E61">A30</font></span></a>, <a class="txt" href="ens_1.html#E10"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<a NAME="E41:53_1"/>
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:53_1"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="ens_1.html#E10"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<a NAME="E42:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A34</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></span>;<br/>
<a NAME="E43:53_1"/>
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:53_1"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="ens_1.html#E10"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/>
<a NAME="E44:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A35</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a></span>;<br/>
<a NAME="E45:53_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:53_1"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="ens_1.html#T28" target="_self" title="ENS_1:th.28">Th28</a></span>;<br/>
<a NAME="E46:53_1"/><span class="kw">then </span>
 <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c12">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c13">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E44:53_1"><span class="lab"><font color="Green" title="E66">A35</font></span></a>, <a class="txt" href="#E30:53_1"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="txt" href="#E42:53_1"><span class="lab"><font color="Green" title="E65">A34</font></span></a>, <a class="txt" href="#E28:53_1"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="txt" href="#E14:53_1"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="txt" href="#E40:53_1"><span class="lab"><font color="Green" title="E64">A33</font></span></a>, <a class="txt" href="#E31:53_1"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="ref" href="cat_1.html#D11" title="CAT_1:def.11">CAT_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E47:53_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E18:53_1"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E17:53_1"><span class="lab"><font color="Green" title="E48">A14</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>
<a NAME="E7:53"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ens_1.html#E17"><span class="lab"><font color="Green" title="E10">Lm3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:53"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ens_1.html#D9" target="_self" title="ENS_1:def.9">ENS_1:def 9</a> <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>
