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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:31"/>
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> implies for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">Z</font>,<font color="Maroon">X</font>  st <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</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="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Z</font>,<font color="Maroon">X</font>, <font color="Maroon">h</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Z</font>,<font color="Maroon">X</font>;<br/>



<a NAME="E2:31_1"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:31_1"/>
( <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font> implies <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T49">FUNCT_1:49</a></i>;<br/>


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

<b>assume </b><a NAME="E4:31"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">Z</font>,<font color="Maroon">X</font>  st <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>let </b><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>assume </b><a NAME="E1:31_2"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">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="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <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> )
 ;<br/><a NAME="E2:31_2"/><b>then </b>
( <font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>,<font color="Maroon">X</font> &amp; <font color="Maroon">h</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>,<font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:31_2"/>
( <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> implies <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#T49">FUNCT_1:49</a></i>;<br/>


</div>
