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

<b>let </b><font color="Maroon" title="c1">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="c1">S</font> holds   <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Olive" title="b1">seq</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">seq</font></span><br/><b>let </b><font color="Maroon" title="c2">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="c1">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> <font color="Maroon" title="c2">seq</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font></span><br/>



<b>set </b><font color="Maroon" title="c3">cseq</font> =  <a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c2">seq</font>,<font color="Maroon" title="c1">S</font>;<br/>
<a NAME="E1:123"/><i><font color="Green" title="E97">A1</font></i>: 
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> <font color="Maroon" title="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <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"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <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="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</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="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<b>assume </b><a NAME="E1:123_1"/><i><font color="Green" title="E98">B1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">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> <font color="Maroon" title="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</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="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:123_1"><i><font color="Green" title="E98">B1</font></i></a></i>;<br/>
<a NAME="E3:123_1"/>
<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="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">x</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="c2">seq</font>,<font color="Maroon" title="c1">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="c5">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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/>
<b>hence </b><a NAME="E4:123_1"/>
<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="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</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="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 ; <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="E2:123"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c2">seq</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K28" title="MODELC_2:func.28">CastSeq</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">CastSeq</a> <font color="Maroon" title="c2">seq</font>,<font color="Maroon" title="c1">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:123"><i><font color="Green" title="E97">A1</font></i></a>, <a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E3:123"/>
 <a href="modelc_2.html#K32" title="MODELC_2:func.32">Shift</a> <font color="Maroon" title="c2">seq</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font>
 <b>by </b><i><a class="ref" href="modelc_2.html#D42" target="_self" title="MODELC_2:def.42">DefCasetSeq2</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>
