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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="funct_1.html#NM1">Function</a>] <b>means </b> ex <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Olive">A</font> : verum </span>}</span>  &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Olive">A</font> : verum </span>}</span>  ) );<br/>
<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>;<br/>

<b>thus </b><a NAME="E1:21_1_2"/>
( <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>] implies <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_2_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>] )
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21_1_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E4:21_1_2_1"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E5:21_1_2_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 ;<br/>
<b>consider </b><font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:21_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E8:21_1_2_1"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E9:21_1_2_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> : verum </span>}</span> 
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:21_1_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 ;<br/><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , , , </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:21_1_2_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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

<b>thus </b><a NAME="E2:21_1_2"/>
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 ;<br/>


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