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<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">h</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font>
 ;<br/>

<i><font color="Green">E33</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">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</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">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</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">f</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">g</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
;<br/>
<i><font color="Green">E34</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">h</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</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">h</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</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">h</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">g</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
;<br/>
<b>then </b><i><font color="Green">E35</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">h</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</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">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</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">f</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">g</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_4"/><i><font color="Green">E36</font></i>: 
<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">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span>
 ;<br/><a NAME="E2:27_4"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></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">g</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_4"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font></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">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="scmfsa_9.html#T2" target="_self">Th2</a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:27_4"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_4_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1_1"><b>suppose </b></a><a NAME="E1:27_4_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="scmfsa_9.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:27_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="scmfsa_9.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1_2"><b>suppose </b></a><a NAME="E1:27_4_1_2"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<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"><font color="Maroon">h</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="scmfsa_9.html#E1:27"><i><font color="Green">E32</font></i></a>, <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"><font color="Maroon">h</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="scmfsa_9.html#E2:27"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:27"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">D</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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