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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">C</font> implies ( <font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> ) )</span><br/>





<a NAME="E1:69"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c3">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T65" target="_self" title="ORDINAL3:th.65">Th78</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:69"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )</span><br/>

<a NAME="E3:69"/><span class="kw">then </span>
<font color="Maroon" title="c3">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T38" title="ORDINAL2:th.38">ORDINAL2:38</a></span>;<br/>
<a NAME="E4:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 ex <font color="Olive" title="b1">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#D6" target="_self" title="ORDINAL3:def.6">Def7</a></span>;<br/>
<a NAME="E5:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T24" target="_self" title="ORDINAL3:th.24">Th27</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:69"/>
(  not <font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> or  not <font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K7" title="ORDINAL3:func.7">mod^</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:69"/><span class="kw">then </span>
<font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="ordinal3.html#K6" title="ORDINAL3:func.6">div^</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="ordinal3.html#T21" target="_self" title="ORDINAL3:th.21">Th24</a>, <a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a>, <a class="ref" href="ordinal2.html#T41" title="ORDINAL2:th.41">ORDINAL2:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:69"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="ordinal1.html#T5" title="ORDINAL1:th.5">ORDINAL1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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