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<b>let </b><font color="Maroon">ADG</font> be   <a href="afvect0.html#NM2">Proper_Uniquely_Two_Divisible_Group</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>;<br/><b>let </b><font color="Maroon">o'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font>;<br/>





<b>assume </b><a NAME="E1:77"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:77_1"/><i><font color="Green">E36</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="E2:77_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/><b>reconsider </b><font color="Maroon">x1'</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2'</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font> <b>by </b><i><a class="ref" href="afvect0.html#T27" target="_self">Th27</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x2'</font> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1'</font>
<b>by </b><i><a class="ref" href="afvect0.html#T26" target="_self">Th26</a>, <a class="ref" href="afvect0.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x1'</font>
<b>by </b><i><a class="ref" href="afvect0.html#T26" target="_self">Th26</a></i>
;<br/><b>hence </b><a NAME="E5:77_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:77"/>
<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#D8">FUNCT_1:def 8</a></i>;<br/>


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