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

<b>consider </b><font color="Maroon">D1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:13_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">D2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">D3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:13_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D3</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>3</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:13_1"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font>
 <b>and </b><br/><a NAME="E9:13_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:13_1"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font>
 <b>and </b><br/><a NAME="E12:13_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon">f3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E14:13_1"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D3</font>
 <b>and </b><br/><a NAME="E15:13_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D3</font> holds <br/><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f3</font>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f3</font>
;<br/>

<i><font color="Green">E79</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f3</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f3</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f3</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
;<br/>
<b>thus </b><a NAME="E17:13_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f3</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:13_1_2_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_1_2"/>
<font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_2"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:13_1_2_2"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:13_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">D1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E4:13_1_2"/>
<font color="Maroon">D3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_3"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:13_1_2_3"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D3</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:13_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f3</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i>, , , , , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

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

<b>assume </b><a NAME="E6:13_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<a NAME="E7:13_1_2"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>] or <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">x</font>] or <font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">x</font>] )
 
<b>by </b><i/>;<br/>
<a NAME="E8:13_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D3</font> )
 
<b>by </b><i>, , , <a class="txt" href="recdef_2.html#E1:13"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E9:13_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D3</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:13_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f3</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

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

<b>assume </b><a NAME="E18:13_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c</font>]
 ;<br/><a NAME="E2:13_1_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:13_1_3"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c</font>]
 <b>by </b><i>, , </i>;<br/><a NAME="E4:13_1_3"/><b>then </b><i><font color="Green">E86</font></i>: 
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 <b>by </b><i/>;<br/><a NAME="E5:13_1_3"/>
<font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c</font>]
 <b>by </b><i>, , </i>;<br/><a NAME="E6:13_1_3"/><b>then </b>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D3</font>
 <b>by </b><i/>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="recdef_2.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c</font>)
<b>by </b><i>, <a class="txt" href="recdef_2.html#E1:13"><i><font color="Green">E30</font></i></a></i>
;<br/><br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">c</font>]
 ;<br/><a NAME="E2:13_1_4"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:13_1_4"/>
<font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c</font>]
 <b>by </b><i>, , </i>;<br/><a NAME="E4:13_1_4"/><b>then </b>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D3</font>
 <b>by </b><i/>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="txt" href="recdef_2.html#E1:13"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c</font>)
<b>by </b><i>, <a class="txt" href="recdef_2.html#E1:13"><i><font color="Green">E30</font></i></a></i>
;<br/><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E21:13_1"/>
<font color="Maroon">P<sub>3</sub></font>[<font color="Maroon">c</font>]
 ;<br/>

<a NAME="E22:13_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D3</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> = 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c</font>)
<b>by </b><i>, </i>
;<br/><br/>


</div>
