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<b>let </b><font color="Maroon" title="c1">ADG</font> be   <a href="tdgroup.html#NM2" title="TDGROUP:NM.2">Uniquely_Two_Divisible_Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font> holds <br/> ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> iff <font color="Olive" title="b1">a</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Olive" title="b3">c</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> iff <font color="Maroon" title="c2">a</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c4">c</font> )</span><br/>









<a NAME="E1:13"/><i><font color="Green" title="E7">A1</font></i>: 
the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <font color="Maroon" title="c1">ADG</font>
 
;<br/>
<div><i><font color="Green" title="E8">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c4">c</font></span><br/><a NAME="E2:13_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <font color="Maroon" title="c1">ADG</font>
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="tdgroup.html#D6" target="_self" title="TDGROUP:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon" title="c2">a</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="tdgroup.html#D4" target="_self" title="TDGROUP:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_2"/>
<font color="Maroon" title="c2">a</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span><br/><a NAME="E2:13_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>
 <b>by </b><i><a class="ref" href="tdgroup.html#D4" target="_self" title="TDGROUP:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E3:13_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="ref" href="tdgroup.html#D6" target="_self" title="TDGROUP:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:13"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> iff <font color="Maroon" title="c2">a</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E8">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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