<?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  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> holds <br/><font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a> </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"> (  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> implies <font color="Maroon" title="c2">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>  )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E2:49"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<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="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"> <font color="Maroon" title="c2">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">b</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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_1.html#D15" title="CAT_1:def.15">CAT_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">b</font>,<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>  &amp;  ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">h</font> being   <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="c4">b</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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">m</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="c4">b</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">h</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E4:49"/><span class="lab"><font color="Green" title="E34">A2</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="c4">b</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">h</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="c4">b</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="#E2:49"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="ens_1.html#T15" target="_self" title="ENS_1:th.15">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:49"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">b</font>,<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="ref" href="ens_1.html#T27" target="_self" title="ENS_1:th.27">Th27</a></span>; <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">b<sub>1</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/>

<a NAME="E6:49"/>
<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="c4">b</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">h</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="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E34">A2</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="c7">f</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="c4">b</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">h</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">f</font>
; <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">b<sub>1</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font>; <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="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font></span><br/>

<a NAME="E8:49"/>
 <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="c4">b</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="c9">m</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="c4">b</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">h</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>, <font color="Maroon" title="c10">m9</font> = <font color="Maroon" title="c8">g</font> 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="#E4:49"><span class="lab"><font color="Green" title="E34">A2</font></span></a></span>;<br/>
<a NAME="E10:49"/>
<font color="Maroon" title="c8">g</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="c4">b</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>;<br/>
<a NAME="E11:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c8">g</font> <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="c4">b</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#T27" target="_self" title="ENS_1:th.27">Th27</a></span>;<br/>
<a NAME="E12:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c10">m9</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#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c4">b</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m9</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#T16" target="_self" title="ENS_1:th.16">Th16</a></span>;<br/>
<a NAME="E13:49"/><span class="kw">then </span>
<font color="Maroon" title="c10">m9</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <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="c4">b</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="#E11:49"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="ens_1.html#E29"><span class="lab"><font color="Green" title="E20">Lm4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:49"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E12:49"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="funct_2.html#T66" title="FUNCT_2:th.66">FUNCT_2:66</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>
