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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E1:18"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
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><span class="lab"><font color="Green" title="E8">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:18_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="tdgroup.html#D6" target="_self" title="TDGROUP:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:18_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tdgroup.html#D4" target="_self" title="TDGROUP:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:18_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tdgroup.html#D4" target="_self" title="TDGROUP:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:18_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tdgroup.html#D6" target="_self" title="TDGROUP:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:18"/>
( <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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