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<b>consider </b><font color="Maroon">ADG</font> being   <a href="tdgroup.html#NM2">Uniquely_Two_Divisible_Group</a><b> such that </b><br/><a NAME="E2:25_1_1"/><i><font color="Green">E19</font></i>: 
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font> st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font>
 <b>by </b><i/>;<br/>
<a NAME="E3:25_1_1"/>
(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span> st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span>  st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">c</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span>  st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">c</font>,<font color="Olive">d</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p</font>,<font color="Olive">q</font> holds <br/><font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span>  ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span> st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">a'</font>, <font color="Olive">b'</font>, <font color="Olive">c'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span>  st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a'</font>,<font color="Olive">b'</font> &amp; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a'</font>,<font color="Olive">c'</font> holds <br/><font color="Olive">b</font>,<font color="Olive">c</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b'</font>,<font color="Olive">c'</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span>  ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span> st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">c</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">b'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span>  st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">c</font> &amp; <font color="Olive">a</font>,<font color="Olive">b'</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b'</font>,<font color="Olive">c</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b'</font> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font></span>)</span>  st <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> holds <br/><font color="Olive">a</font>,<font color="Olive">c</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">d</font> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:25_1_1"/><b>then </b>
( not  <a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font> is <a href="realset2.html#V3">trivial</a> &amp;  <a href="tdgroup.html#K2" target="_self">AV</a> <font color="Maroon">ADG</font> is <a href="tdgroup.html#V2" target="_self">AffVect-like</a> )
 
<b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="analoaf.html#L1">AffinStruct</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="analoaf.html#V1">strict</a> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tdgroup.html#V2" target="_self">AffVect-like</a> )
 ;<br/>


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