<?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="E1:90"/>
( ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">y</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> ) implies <font color="Maroon">f</font> is <a href="funct_1.html#V2" target="_self">one-to-one</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:90_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">y</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" target="_self">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E2:90_1"/><i><font color="Green">E39</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> )
 ;<br/>

<a NAME="E3:90_1"/><b>then </b><i><font color="Green">E40</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#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:90_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="funct_1.html#E2:90_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:90_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="funct_1.html#E2:90_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="funct_1.html#E5:90_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E8:90_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x2</font></span>)</span></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="E9:90_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="funct_1.html#E3:90_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="funct_1.html#E7:90_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="funct_1.html#E2:90"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<a NAME="E10:90_1"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:90_1"/>
(  not <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> or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="txt" href="funct_1.html#E7:90_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="funct_1.html#E2:90"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/>


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

<b>assume </b><a NAME="E2:90"/><i><font color="Green">E43</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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:90"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:90"/><i><font color="Green">E44</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">f</font>
 <b>and </b><br/><a NAME="E6:90"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<a NAME="E7:90"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> iff <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:90_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:90_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:90_2_1"/><b>then </b>
( <font color="Maroon">z</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="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:90_2_1"/><b>then </b>
( <font color="Maroon">z</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="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:90_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/>


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

<a NAME="E2:90_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></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/>
<b>hence </b><a NAME="E3:90_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:90"/>
<font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></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/>


</div>
