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<b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</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">k11</font>, <font color="Olive" title="b2">k21</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k1</font> &amp; <font color="Olive" title="b2">k21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k2</font> holds <br/><font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="card_fin.html#K3" title="CARD_FIN:func.3">|</a> <font color="Olive" title="b2">k21</font></span>)</span> <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k11</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) &amp; ( for <font color="Olive" title="b1">k11</font>, <font color="Olive" title="b2">k21</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k1</font> &amp; <font color="Olive" title="b2">k21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k2</font> holds <br/><font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="card_fin.html#K3" title="CARD_FIN:func.3">|</a> <font color="Olive" title="b2">k21</font></span>)</span> <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k11</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) implies <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> )</span><br/>

<b>assume </b><a NAME="E1:31_1_2"/><i><font color="Green" title="E22">B1</font></i>: 
( ( for <font color="Olive" title="b1">k11</font>, <font color="Olive" title="b2">k21</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k1</font> &amp; <font color="Olive" title="b2">k21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k2</font> holds <br/><font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="card_fin.html#K3" title="CARD_FIN:func.3">|</a> <font color="Olive" title="b2">k21</font></span>)</span> <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k11</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) &amp; ( for <font color="Olive" title="b1">k11</font>, <font color="Olive" title="b2">k21</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k1</font> &amp; <font color="Olive" title="b2">k21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k2</font> holds <br/><font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="card_fin.html#K3" title="CARD_FIN:func.3">|</a> <font color="Olive" title="b2">k21</font></span>)</span> <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k11</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">k12</font> = <font color="Maroon" title="c2">k1</font>, <font color="Maroon" title="c7">k22</font> = <font color="Maroon" title="c3">k2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:31_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="card_fin.html#K3" title="CARD_FIN:func.3">|</a> <font color="Maroon" title="c7">k22</font></span>)</span> <a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k12</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:31_1_2"><i><font color="Green" title="E22">B1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:31_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="#E1:31_1_2"><i><font color="Green" title="E22">B1</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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