<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 1 <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="Olive" title="b1">s</font> holds <br/> <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">n</font>)</span>)</span>,<font color="Olive" title="b3">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">m</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">s</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 1 <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">s</font> holds <br/> <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Olive" title="b1">n</font>)</span>)</span>,<font color="Olive" title="b2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">m</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <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">s</font> implies  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>) )</span><br/>





<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
1 <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">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span><br/>

<a NAME="E2:19"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</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">s</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">s</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D3" target="_self" onmouseover="rs('descip_1/D3')" onmouseout="rh()">Def3</a>, <a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/>
<a NAME="E3:19"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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">s</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">s</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D3" target="_self" onmouseover="rs('descip_1/D3')" onmouseout="rh()">Def3</a>, <a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/>
<a NAME="E4:19"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</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> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="descip_1.html#D3" target="_self" onmouseover="rs('descip_1/D3')" onmouseout="rh()">Def3</a>, <a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E10">A2</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/><a NAME="E2:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E3:19_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">i1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</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">s</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a>, <a class="ref" href="int_1.html#T57" onmouseover="rs('int_1/T57')" onmouseout="rh()">INT_1:57</a></span>;<br/><a NAME="E5:19_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E6:19_1"/>
<font color="Maroon" title="c6">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="int_1.html#T58" onmouseover="rs('int_1/T58')" onmouseout="rh()">INT_1:58</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E7:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c6">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>;<br/><a NAME="E8:19_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c6">i1</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</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">s</font></span>)</span> <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="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</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">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T65" onmouseover="rs('nat_d/T65')" onmouseout="rh()">NAT_D:65</a></span>;<br/><span class="lab"><font color="Green" title="E19">A11</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</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">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</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">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:19_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</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">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>
;<br/><span class="lab"><font color="Green" title="E20">A12</font></span>: <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="descip_1.html#D3" target="_self" onmouseover="rs('descip_1/D3')" onmouseout="rh()">Def3</a>, <a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E10">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="descip_1.html#D3" target="_self" onmouseover="rs('descip_1/D3')" onmouseout="rh()">Def3</a>, <a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:19_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a></span>
;<br/><a NAME="E11:19_1"/>
<span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="descip_1.html#D3" target="_self" onmouseover="rs('descip_1/D3')" onmouseout="rh()">Def3</a>, <a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:19_1"/>
<span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p2">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:19"/>
 <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c2">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="finseq_2.html#T9" onmouseover="rs('finseq_2/T9')" onmouseout="rh()">FINSEQ_2:9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
