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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</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">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#K3" title="AFINSQ_2:func.3">mid</a> <font color="Maroon" title="c1">p</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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K3" title="AFINSQ_2:func.3">mid</a> <font color="Maroon" title="c1">p</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/>





<span class="kw">set </span><font color="Maroon" title="c4">k11</font> = <font color="Maroon" title="c2">k1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">k21</font> = <font color="Maroon" title="c3">k2</font>;<br/>
<a NAME="E1:37"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">k2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T59" title="AFINSQ_1:th.59">AFINSQ_1:59</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:37"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K3" title="AFINSQ_2:func.3">mid</a> <font color="Maroon" title="c1">p</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/>

<a NAME="E3:37"/><span class="kw">then </span>
<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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<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="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E5:37"/>
<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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E6:37"/><span class="kw">then </span>
<font color="Maroon" title="c2">k1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:37"/>
 <a href="afinsq_2.html#K3" title="AFINSQ_2:func.3">mid</a> <font color="Maroon" title="c1">p</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="afinsq_2.html#T15" target="_self" title="AFINSQ_2:th.15">Th15</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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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