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

<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">seq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <font color="Olive" title="b1">S</font> holds   <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Olive" title="b2">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Olive" title="b2">seq</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <font color="Maroon" title="c3">S</font> holds   <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Olive" title="b1">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Olive" title="b1">seq</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">seq</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><br/>







<b>set </b><font color="Maroon" title="c5">nk</font> = <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font>;<br/>
<b>set </b><font color="Maroon" title="c6">t1</font> =  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font>;<br/>
<b>set </b><font color="Maroon" title="c7">t2</font> =  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c2">n</font>;<br/>
<b>set </b><font color="Maroon" title="c8">t3</font> =  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">cseq</font> =  <a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font>;<br/>
<b>set </b><font color="Maroon" title="c10">ct1</font> =  <a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font>;<br/>
<a NAME="E1:124"/><i><font color="Green" title="E98">A4</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="124_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span><br/>

<a NAME="E1:124_1"/>
<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font>
 
<b>by </b><i><a class="ref" href="modelc_2.html#D42" target="_self" title="MODELC_2:def.42">DefCasetSeq2</a></i>;<br/>
<b>hence </b><a NAME="E2:124_1"/>
<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="modelc_2.html#D43" target="_self" title="MODELC_2:def.43">DefShift01</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:124"/><i><font color="Green" title="E99">A5</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="124_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font></span><br/>

<span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font> = 
<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="ref" href="modelc_2.html#D43" target="_self" title="MODELC_2:def.43">DefShift01</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:124"><i><font color="Green" title="E98">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E2:124_2"/>
<span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#D43" target="_self" title="MODELC_2:def.43">DefShift01</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:124"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <span class="p3">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c3">S</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K30" title="MODELC_2:func.30">Shift</a> <span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="#E2:124"><i><font color="Green" title="E99">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:124"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c4">seq</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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