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<span class="kw">let </span><font color="Maroon" title="c1">s</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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="rewrite2.html#V1" title="REWRITE2:attr.1">XFinSequence-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Olive" title="b2">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="rewrite2.html#V1" title="REWRITE2:attr.1">XFinSequence-yielding</a>  <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">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> )</span><br/>





<div><span class="lab"><font color="Green" title="E9">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">k</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="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/><a NAME="E2:18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:18_1_1_1"/>
( <font color="Maroon" title="c4">k</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">p</font> or  not <font color="Maroon" title="c4">k</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">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:18_1_1_1_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c4">k</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">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/><div class="add"><a NAME="E2:18_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<font color="Maroon" title="c1">s</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c1">s</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_1_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:18_1_1_1_2"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
not <font color="Maroon" title="c4">k</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">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/><div class="add"><a NAME="E2:18_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:18_1_1_1_2"/>
( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1_1_2"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <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">consider </span><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:18_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
<font color="Maroon" title="c4">k</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">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E6:18_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c5">i</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="c3">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E2:18_1_1_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="rewrite2.html#T2" target="_self" title="REWRITE2:th.2">Th2</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:18_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c5">i</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="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18_1_1_1_2"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:18_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<font color="Maroon" title="c5">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>;<br/><a NAME="E9:18_1_1_1_2"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<font color="Maroon" title="c1">s</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:18_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c1">s</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_1_1_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_1_1_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="rewrite2.html#D2" target="_self" title="REWRITE2:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E8:18_1_1_1_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="txt" href="#E9:18_1_1_1_2"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:18_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
;<br/>
<a NAME="E3:18"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:18"/>
<font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="rewrite2.html#K2" title="REWRITE2:func.2">^+</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E10">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">k</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="c4">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:18_3"/><span class="lab"><font color="Green" title="E11">A14</font></span>: 
<font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/><a NAME="E2:18_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A15</font></span>: 
<font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:18_3_1_1"/>
( <font color="Maroon" title="c4">k</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">p</font> or  not <font color="Maroon" title="c4">k</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">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:18_3_1_1_1"/><span class="lab"><font color="Green" title="E13">A16</font></span>: 
<font color="Maroon" title="c4">k</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">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/><div class="add"><a NAME="E2:18_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A17</font></span>: 
<font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c1">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_3"><span class="lab"><font color="Green" title="E12">A15</font></span></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c1">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_3_1_1_1"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_3_1_1_1"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_3_1_1_1"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:18_3_1_1_2"/><span class="lab"><font color="Green" title="E13">A18</font></span>: 
not <font color="Maroon" title="c4">k</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">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/><div class="add"><a NAME="E2:18_3_1_1_2"/><span class="lab"><font color="Green" title="E14">A19</font></span>: 
<font color="Maroon" title="c4">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_3"><span class="lab"><font color="Green" title="E12">A15</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:18_3_1_1_2"/>
( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_3_1_1_2"><span class="lab"><font color="Green" title="E13">A18</font></span></a>, <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">consider </span><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:18_3_1_1_2"/><span class="lab"><font color="Green" title="E15">A20</font></span>: 
<font color="Maroon" title="c4">k</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">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E6:18_3_1_1_2"/><span class="lab"><font color="Green" title="E16">A21</font></span>: 
( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c5">i</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="c3">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_3"><span class="lab"><font color="Green" title="E11">A14</font></span></a>, <a class="txt" href="#E2:18_3_1_1_2"><span class="lab"><font color="Green" title="E14">A19</font></span></a>, <a class="ref" href="rewrite2.html#T2" target="_self" title="REWRITE2:th.2">Th2</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:18_3_1_1_2"/><span class="lab"><font color="Green" title="E17">A22</font></span>: 
<font color="Maroon" title="c5">i</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="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18_3_1_1_2"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:18_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A23</font></span>: 
<font color="Maroon" title="c5">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>;<br/><a NAME="E9:18_3_1_1_2"/><span class="lab"><font color="Green" title="E19">A24</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c1">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_3"><span class="lab"><font color="Green" title="E12">A15</font></span></a>, <a class="txt" href="#E5:18_3_1_1_2"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c1">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_3_1_1_2"><span class="lab"><font color="Green" title="E17">A22</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_3_1_1_2"><span class="lab"><font color="Green" title="E17">A22</font></span></a>, <a class="ref" href="rewrite2.html#D3" target="_self" title="REWRITE2:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_3_1_1_2"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="txt" href="#E8:18_3_1_1_2"><span class="lab"><font color="Green" title="E18">A23</font></span></a>, <a class="txt" href="#E9:18_3_1_1_2"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:18_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="rewrite2.html#K1" title="REWRITE2:func.1">.</a> <font color="Maroon" title="c4">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T5" target="_self" title="REWRITE2:th.5">Th5</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
;<br/>
<a NAME="E7:18"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="rewrite2.html#K3" title="REWRITE2:func.3">+^</a> <font color="Maroon" title="c1">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E10">A13</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</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/>


</div>
