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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Olive" title="b1">S</font> holds   <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K29" title="MODELC_2:func.29">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c3">S</font> holds   <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K29" title="MODELC_2:func.29">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/><span class="kw">let </span><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#K25" title="MODELC_2:func.25">Inf_seq</a> <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K29" title="MODELC_2:func.29">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/>







<span class="kw">set </span><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/>
<span class="kw">set </span><font color="Maroon" title="c6">t1</font> =  <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c1">k</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">cseq</font> =  <a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">ct1</font> =  <a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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="E1:128"/><span class="lab"><font color="Green" title="E97">A1</font></span>: 
 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#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K27" title="MODELC_2:func.27">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="128_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">m</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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">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="c9">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="E1:128_1"/>
<span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <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="c9">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_2.html#D41" target="_self" title="MODELC_2:def.41">Def44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:128_1"/>
<span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">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="c9">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>; <a class="txt" 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>
<a NAME="E2:128"/>
 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"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p3">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K8" title="NAT_1:func.8">^\</a> <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"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</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> <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="128_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">m</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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p3">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K8" title="NAT_1:func.8">^\</a> <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="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</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> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p3">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K8" title="NAT_1:func.8">^\</a> <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="c9">m</font> = 
<span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p2">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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="c9">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">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="c9">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E97">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">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="c9">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/>
<span class="kw">hence </span><a NAME="E2:128_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p3">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K8" title="NAT_1:func.8">^\</a> <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="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</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> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>; <a class="txt" 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>
<a NAME="E3:128"/><span class="kw">then </span>
 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"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<span class="p3">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K8" title="NAT_1:func.8">^\</a> <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"><span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">CastSeq</a> (<font color="Maroon" title="c4">seq</font>,<font color="Maroon" title="c3">S</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:128"/>
 <a href="modelc_2.html#K29" title="MODELC_2:func.29">Shift</a> (<span class="p1">(<span class="default"><a href="modelc_2.html#K29" title="MODELC_2:func.29">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#K29" title="MODELC_2:func.29">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 class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
