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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>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:78"/><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/>

<b>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>;<br/>
<a NAME="E2:78"/><i><font color="Green">E36</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> <a href="hidden.html#R1">=</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">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:78"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tdgroup.html#T9">TDGROUP:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> ;<br/><b>set </b><font color="Maroon">x'</font> = <font color="Maroon">y'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">o'</font>;<br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">o'</font></span>)</span> = 
<font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">o'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="afvect0.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">o'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">ADG</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E4:78_1"/>
<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>
 <b>by </b><i><a class="ref" href="afvect0.html#T28" target="_self">Th28</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:78"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:78"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="afvect0.html#T28" target="_self">Th28</a>, <a class="ref" href="afvect0.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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