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<b>let </b><font color="Maroon" title="c1">f</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">k1</font>, <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">k2</font> holds <br/> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">mid</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><b>let </b><font color="Maroon" title="c2">k1</font>, <font color="Maroon" title="c3">k2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k2</font> implies  <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">mid</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">mid</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<b>reconsider </b><font color="Maroon" title="c4">k11</font> = <font color="Maroon" title="c2">k1</font>, <font color="Maroon" title="c5">k21</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="E3:32"/><i><font color="Green" title="E25">A2</font></i>: 
 <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">mid</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</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="c5">k21</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="c4">k11</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="afinsq_2.html#D3" target="_self" title="AFINSQ_2:def.3">AB540b</a></i>;<br/>
<a NAME="E4:32"/>
<font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:32"/><b>then </b><i><font color="Green" title="E26">A17</font></i>: 
<font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/>
<a NAME="E6:32"/>
<font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:32"/><b>then </b><i><font color="Green" title="E27">A18</font></i>: 
<font color="Maroon" title="c2">k1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:32"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</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="c5">k21</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k21</font>
 
<b>by </b><i><a class="ref" href="afinsq_2.html#T13" target="_self" title="AFINSQ_2:th.13">TH80f</a></i>;<br/>
<b>hence </b><a NAME="E9:32"/>
 <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">mid</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E3:32"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="afinsq_2.html#T15" target="_self" title="AFINSQ_2:th.15">TH80e</a>, <a class="txt" href="#E5:32"><i><font color="Green" title="E26">A17</font></i></a>, <a class="txt" href="#E7:32"><i><font color="Green" title="E27">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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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