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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2" target="_self">one-to-one</a>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:54"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:54"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>

<a NAME="E5:54"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="funct_1.html#E1:54"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E6:54"/>
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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><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="E1:54_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:54_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="funct_1.html#E2:54"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:54_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="funct_1.html#E4:54"><i><font color="Green">E28</font></i></a>, <a class="txt" href="funct_1.html#E1:54_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:54_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="funct_1.html#E1:54"><i><font color="Green">E25</font></i></a>, <a class="txt" href="funct_1.html#E2:54_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T54" target="_self">Th54</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:54"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K2" target="_self">"</a> 
 <b>by </b><i><a class="txt" href="funct_1.html#E3:54"><i><font color="Green">E27</font></i></a>, <a class="txt" href="funct_1.html#E5:54"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T9" target="_self">Th9</a></i>;<br/>


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