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<span class="kw">let </span><font color="Maroon" title="c1">a</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"> for <font color="Olive" title="b1">fs</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">fs</font> holds <br/> ex <font color="Olive" title="b2">fs1</font>, <font color="Olive" title="b3">fs2</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">fs1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b3">fs2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">fs2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">fs</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">fs</font> implies  ex <font color="Olive" title="b1">fs1</font>, <font color="Olive" title="b2">fs2</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Maroon" title="c2">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">fs1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">fs2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fs</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">fs</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">fs1</font>, <font color="Olive" title="b2">fs2</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Maroon" title="c2">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">fs1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">fs2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fs</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> )</span><br/>

<a NAME="E2:40"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fs</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">b</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fs</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c4">d</font> = <font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/>
<a NAME="E4:40"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">b</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">fs3</font>, <font color="Maroon" title="c6">fs2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E6:40"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">fs3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">and </span><br/><a NAME="E7:40"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">fs2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E8:40"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c2">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fs3</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">fs2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T25" title="FINSEQ_2:th.25">FINSEQ_2:25</a></span>;<br/>
<a NAME="E9:40"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c8">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:40"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c5">fs3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">v</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="finseq_2.html#T21" title="FINSEQ_2:th.21">FINSEQ_2:21</a></span>;<br/>
<a NAME="E12:40"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">fs1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></span>;<br/>
<a NAME="E13:40"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E14:40"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">fs3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<a NAME="E15:40"/><span class="kw">then </span>
<font color="Maroon" title="c5">fs3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E16:40"/><span class="kw">then </span>
<font color="Maroon" title="c2">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:40"/>
 ex <font color="Olive" title="b1">fs1</font>, <font color="Olive" title="b2">fs2</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Maroon" title="c2">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">fs1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">fs2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fs</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E12:40"><span class="lab"><font color="Green" title="E30">A6</font></span></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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