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

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>thus </b><a NAME="E1:38"/>
( <font color="Maroon">f</font> is <a href="funct_1.html#V2" target="_self">one-to-one</a> implies for <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">h</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">h</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">h</font> holds <br/><font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2" target="_self">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:38_1"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:38_1"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:38_1"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E5:38_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font>
 ;<br/>

<a NAME="E6:38_1"/>
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>  <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" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </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="E1:38_1_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>
 ;<br/>

<a NAME="E2:38_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> &amp; ( <font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> ) &amp; ( <font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> implies <font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> ) )
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E3:38_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:38_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:38"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">h</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">h</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">h</font> holds <br/><font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font>
 ;<br/>

<a NAME="E3:38"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:38_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E2:38_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x2</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">x1</font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:38_2"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E5:38_2"/><i><font color="Green">E47</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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" target="_self">FUNCT_1:sch 3</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">x2</font>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:38_2"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E8:38_2"/><i><font color="Green">E49</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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" target="_self">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E9:38_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:38_2"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:38_2"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:38_2"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E13:38_2"/><b>then </b><i><font color="Green">E69</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E14:38_2"/>
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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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> ;<br/>

<b>assume </b><a NAME="E1:38_2_1"/>
<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"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E2:38_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E3:38_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E15:38_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E16:38_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:38"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2" target="_self">one-to-one</a>
 <b>by </b><i/>;<br/>


</div>
