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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<i><font color="Green">E15</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>
;<br/>
<a NAME="E2:26"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1"><b>suppose </b></a><a NAME="E1:26_2_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_2_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="funct_4.html#E1:26_2_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_4.html#T14" target="_self">Th14</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="funct_4.html#E1:26_2_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="funct_4.html#E1:26_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_2"><b>suppose </b></a><a NAME="E1:26_2_1_1_2"/><i><font color="Green">E17</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_2_1_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="funct_4.html#E1:26_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:26_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="funct_4.html#E1:26_2_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="funct_4.html#E2:26_2_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="funct_4.html#E1:26_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26"/>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_4.html#E1:26"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
