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

<b>set </b><font color="Maroon">H</font> = <font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font>;<br/>
<b>set </b><font color="Maroon">G</font> = <font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font>;<br/>
<a NAME="E1:6_1_1"/><i><font color="Green">E29</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:6_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <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="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A0</font> = <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> ;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A0</font>;<br/><b>thus </b><a NAME="E1:6_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> implies <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>thus </b><a NAME="E2:6_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> implies <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:6_1_1_1_1"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A0</font>;<br/><b>thus </b><a NAME="E1:6_1_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> implies <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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>thus </b><a NAME="E2:6_1_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> implies <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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font>
 ;<br/>

<a NAME="E2:6_1_1_2_1"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_1_2_1"/>
<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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:6_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <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="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E7:6_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K2">rng</a> <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="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F0</font> = <font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>,<font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>,<font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A0</font>;<br/><div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_1_3_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K5">\+\</a> <font color="Maroon">A2</font>
 ;<br/><div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_1_3_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:6_1_1_3_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>thus </b><font color="Maroon">F0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="tmap_1.html#T1" target="_self">Th1</a></i>
;<br/></div><b>end;</b></div><div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_1_3_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font>
 ;<br/><a NAME="E2:6_1_1_3_1_2"/>
<font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>hence </b><font color="Maroon">F0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="tmap_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="tmap_1.html#T1" target="_self">Th1</a></i>
;<br/><br/></div><b>end;</b></div><b>thus </b><a NAME="E4:6_1_1_3_1"/>
<font color="Maroon">F0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_1_3_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:6_1_1_3_2"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><i><font color="Green">E53</font></i>: <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E4:6_1_1_3_2"/>
( <font color="Maroon">F0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:6_1_1_3_2"/>
<font color="Maroon">F0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><a NAME="E3:6_1_1_3"/>
( <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K5">\+\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A0</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T93">XBOOLE_1:93</a></i>;<br/><b>hence </b><a NAME="E4:6_1_1_3"/>
<font color="Maroon">F0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <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>take </b>
<font color="Maroon">F0</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_4"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:6_1_1_4"/>
<font color="Maroon">A1</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A0</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A0</font>;<br/><b>assume </b><a NAME="E2:6_1_1_4"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><b>thus </b><font color="Maroon">F0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:6_1_1"/>
<font color="Maroon">F0</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font>
 <b>by </b><i/>;<br/>

<a NAME="E13:6_1_1"/>
( <font color="Maroon">A2</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A0</font> &amp; ( for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A0</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> holds <br/><font color="Maroon">F0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> ) )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E14:6_1_1"/>
<font color="Maroon">F0</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i/>;<br/>


</div>
