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

<b>let </b><font color="Maroon" title="c1">X</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> ;<br/><b>let </b><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E19">A1</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>;<br/>



<b>assume </b><a NAME="E2:23"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1"/>
( <font color="Maroon" title="c3">o2</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>  or <font color="Maroon" title="c3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>suppose </b></a><a NAME="E1:23_1_1"/>
<font color="Maroon" title="c3">o2</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> 
 ;<br/><div class="add"><a NAME="E2:23_1_1"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><a NAME="E3:23_1_1"/>
( <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V4" title="ALTCAT_3:attr.4">mono</a> implies <font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V4" title="ALTCAT_3:attr.4">mono</a>
 ;<br/>

<b>assume </b><a NAME="E2:23_1_1_1"/>
not <font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23_1_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">F</font> )
 <b>and </b><br/><a NAME="E5:23_1_1_1"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
 <b>and </b><br/><a NAME="E6:23_1_1_1"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">o</font> = <font color="Maroon" title="c2">o1</font>;<br/>
<a NAME="E7:23_1_1_1"/><i><font color="Green" title="E26">A8</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="altcat_1.html#T23" title="ALTCAT_1:th.23">ALTCAT_1:23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font>;<br/>
<a NAME="E8:23_1_1_1"/><i><font color="Green" title="E27">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o1</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">C</font> = <font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font>;<br/>
<a NAME="E9:23_1_1_1"/><i><font color="Green" title="E28">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o1</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">o'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">o1</font>;<br/>
<a NAME="E11:23_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E12:23_1_1_1"/><b>then </b><i><font color="Green" title="E29">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o'</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="altcat_3.html#E6:23_1_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E13:23_1_1_1"/><i><font color="Green" title="E30">A12</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">o1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_1"/>
<font color="Maroon" title="c12">y</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"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_1_1"/><i><font color="Green" title="E31">A13</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E4:23_1_1_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o1</font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="altcat_3.html#E3:23_1_1_1_1"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:23_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c2">o1</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E8:23_1_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">B1</font> = <font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c2">o1</font> <b>by </b><i><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></i>;<br/>
<a NAME="E16:23_1_1_1"/><i><font color="Green" title="E31">A14</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">o1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2"/>
<font color="Maroon" title="c13">y</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"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_1_2"/><i><font color="Green" title="E32">A15</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E4:23_1_1_1_2"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o1</font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="altcat_3.html#E9:23_1_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="altcat_3.html#E3:23_1_1_1_2"><i><font color="Green" title="E32">A15</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:23_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c2">o1</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E9:23_1_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">C1</font> = <font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c2">o1</font> <b>by </b><i><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></i>;<br/>
<a NAME="E19:23_1_1_1"/><i><font color="Green" title="E32">A16</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o1</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="altcat_3.html#E13:23_1_1_1"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E20:23_1_1_1"/><i><font color="Green" title="E33">A17</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o1</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="altcat_3.html#E9:23_1_1_1"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="altcat_3.html#E16:23_1_1_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_1_3"/><i><font color="Green" title="E34">A18</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o1</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z</font> = 
<font color="Maroon" title="c5">F</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"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E19:23_1_1_1"><i><font color="Green" title="E32">A16</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E5:23_1_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_1_3"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</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"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23_1_1_1_3"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E20:23_1_1_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_1_3"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E22:23_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E19:23_1_1_1"><i><font color="Green" title="E32">A16</font></i></a>, <a class="txt" href="altcat_3.html#E20:23_1_1_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>then </b><font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">B1</font> = 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o1</font> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c7">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="altcat_3.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="altcat_3.html#E7:23_1_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></i>
<br/>.= 
<font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">C1</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="altcat_3.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="altcat_3.html#E7:23_1_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></i>
;<br/>
<b>hence </b><a NAME="E24:23_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:23_1_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="altcat_3.html#E7:23_1_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="altcat_3.html#E12:23_1_1_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="altcat_3.html#D7" target="_self" title="ALTCAT_3:def.7">Def7</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:23_1_1"/>
( <font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V4" title="ALTCAT_3:attr.4">mono</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1_2"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D7" target="_self" title="ALTCAT_3:def.7">ALTCAT_3:def 7</a><br/>

<b>assume </b><a NAME="E2:23_1_1_2"/><i><font color="Green" title="E23">A20</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c8">C</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">o1</font>;<br/>

<b>assume </b><a NAME="E3:23_1_1_2"/><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">C</font>
 ;<br/>

<a NAME="E4:23_1_1_2"/><i><font color="Green" title="E25">A22</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">o1</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">B1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:23_1_1_2"/><i><font color="Green" title="E26">A23</font></i>: 
( <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">o</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">o1</font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">C1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:23_1_1_2"/><i><font color="Green" title="E27">A24</font></i>: 
( <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">o</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">o1</font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_2"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:23_1_1_2"/><i><font color="Green" title="E28">A25</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E29">A26</font></i>: <font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">B1</font> = 
<font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">C</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="altcat_3.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="altcat_3.html#E3:23_1_1_2"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green" title="E26">A23</font></i></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">C1</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="altcat_3.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="altcat_3.html#E9:23_1_1_2"><i><font color="Green" title="E28">A25</font></i></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_2_2"/><i><font color="Green" title="E30">A27</font></i>: 
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">o</font>
 ;<br/><a NAME="E2:23_1_1_2_2"/><b>then </b><i><font color="Green" title="E31">A28</font></i>: 
( <font color="Maroon" title="c9">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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> <font color="Maroon" title="c9">B1</font> &amp; <font color="Maroon" title="c10">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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> <font color="Maroon" title="c10">C1</font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green" title="E27">A24</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:23_1_1_2_2"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">B1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_2_2"><i><font color="Green" title="E30">A27</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E4:23_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="altcat_3.html#E10:23_1_1_2"><i><font color="Green" title="E29">A26</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_2_2"><i><font color="Green" title="E30">A27</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E5:23_1_1_2_2"/>
<font color="Maroon" title="c9">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="altcat_3.html#E2:23_1_1_2_2"><i><font color="Green" title="E31">A28</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:23_1_1_2"/>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green" title="E27">A24</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>suppose </b></a><a NAME="E1:23_1_2"/><i><font color="Green" title="E21">A29</font></i>: 
<font color="Maroon" title="c3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green" title="E22">A30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_1_1"/>
( <font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">o1</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_2_1_1_1"/>
<font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#T57" title="PARTFUN1:th.57">PARTFUN1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_1_1_2"/>
<font color="Maroon" title="c2">o1</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> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:23_1_2"><i><font color="Green" title="E21">A29</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>thus </b><a NAME="E3:23_1_2"/>
( <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V4" title="ALTCAT_3:attr.4">mono</a> implies <font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_2_2"/>
<font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V4" title="ALTCAT_3:attr.4">mono</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">F1</font> = <font color="Maroon" title="c5">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="ref" href="funct_2.html#T55" title="FUNCT_2:th.55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<a NAME="E3:23_1_2_2"/>
<font color="Maroon" title="c6">F1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#T59" title="PARTFUN1:th.59">PARTFUN1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_2_2"/>
<font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:23_1_2"/>
( <font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V4" title="ALTCAT_3:attr.4">mono</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_2_3"/>
<font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D7" target="_self" title="ALTCAT_3:def.7">ALTCAT_3:def 7</a><br/>

<b>assume </b><a NAME="E2:23_1_2_3"/><i><font color="Green" title="E23">A31</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c8">C</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">o1</font>;<br/>

<b>assume </b><a NAME="E3:23_1_2_3"/>
<font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">C</font>
 ;<br/>

<a NAME="E4:23_1_2_3"/><i><font color="Green" title="E24">A32</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">o1</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">B1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:23_1_2_3"/><i><font color="Green" title="E25">A33</font></i>: 
( <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">o</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">o1</font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_2_3"><i><font color="Green" title="E23">A31</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">C1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:23_1_2_3"/><i><font color="Green" title="E26">A34</font></i>: 
( <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">o</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">o1</font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_2_3"><i><font color="Green" title="E23">A31</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_2_3"><i><font color="Green" title="E24">A32</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:23_1_2_3"/><i><font color="Green" title="E27">A35</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>;<br/>
<b>consider </b><font color="Maroon" title="c12">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:23_1_2_3"/><i><font color="Green" title="E28">A36</font></i>: 
( <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">o2</font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="altcat_3.html#E9:23_1_2_3"><i><font color="Green" title="E27">A35</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E13:23_1_2_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23_1_2"><i><font color="Green" title="E21">A29</font></i></a>, <a class="txt" href="altcat_3.html#E12:23_1_2_3"><i><font color="Green" title="E28">A36</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E14:23_1_2_3"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E15:23_1_2_3"/><b>then </b><i><font color="Green" title="E29">A37</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_2_3"><i><font color="Green" title="E25">A33</font></i></a>, <a class="txt" href="altcat_3.html#E12:23_1_2_3"><i><font color="Green" title="E28">A36</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E16:23_1_2_3"/><b>then </b><i><font color="Green" title="E30">A38</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<font color="Maroon" title="c9">B1</font> = 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_3.html#E15:23_1_2_3"><i><font color="Green" title="E29">A37</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>
<br/>.= 
<font color="Maroon" title="c10">C1</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_2_3"><i><font color="Green" title="E25">A33</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_2_3"><i><font color="Green" title="E26">A34</font></i></a>, <a class="txt" href="altcat_3.html#E16:23_1_2_3"><i><font color="Green" title="E30">A38</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>
;<br/>
<b>hence </b><a NAME="E18:23_1_2_3"/>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_2_3"><i><font color="Green" title="E25">A33</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_2_3"><i><font color="Green" title="E26">A34</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
