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<span class="kw">consider </span><font color="Maroon" title="c4">r</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E2:27_1_1"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D17" title="GROUP_10:def.17">GROUP_10:def 17</a></span>;<br/>
<a NAME="E3:27_1_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c3">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T147" title="GROUP_2:th.147">GROUP_2:147</a></span>;<br/>
<a NAME="E4:27_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T27" title="GROUP_6:th.27">GROUP_6:27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">n</font> =  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">N</font></span>)</span> as   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E6:27_1_1"/>
<font color="Maroon" title="c5">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:27_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<a NAME="E7:27_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r1</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#T2" target="_self" title="GROUPP_1:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:27_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">N</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D17" title="GROUP_10:def.17">GROUP_10:def 17</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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