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

<b>let </b><font color="Maroon" title="c1">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">^omega</a> </span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">l</font>, <font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>









<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">l</font>
 ;<br/>

<a NAME="E2:34"/><i><font color="Green" title="E29">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">^omega</a> <b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">b</font> )
 <b>by </b><i><a class="ref" href="flang_1.html#D1" title="FLANG_1:def.1">FLANG_1:def 1</a></i>;<br/>
<a NAME="E4:34_1"/>
<font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="flang_3.html#T6" target="_self" title="FLANG_3:th.6">Th6</a></i>;<br/>
<a NAME="E5:34_1"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="flang_3.html#E3:34_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="flang_1.html#D1" title="FLANG_1:def.1">FLANG_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:34_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="flang_3.html#E3:34_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="flang_3.html#T18" target="_self" title="FLANG_3:th.18">Th18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:34"/>
<font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:34_2"/><i><font color="Green" title="E30">A4</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="ref" href="flang_3.html#T2" target="_self" title="FLANG_3:th.2">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:34_2"/><i><font color="Green" title="E31">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="flang_3.html#E3:34_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<a NAME="E6:34_2"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="flang_3.html#E5:34_2"><i><font color="Green" title="E31">A5</font></i></a></i>;<br/>
<a NAME="E7:34_2"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="flang_3.html#E3:34_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="flang_1.html#T34" title="FLANG_1:th.34">FLANG_1:34</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">^omega</a> <b> such that </b><br/><a NAME="E9:34_2"/><i><font color="Green" title="E32">A6</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="ref" href="flang_1.html#D1" title="FLANG_1:def.1">FLANG_1:def 1</a></i>;<br/>
<a NAME="E10:34_2"/><i><font color="Green" title="E33">A7</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</font>
 
<b>by </b><i><a class="txt" href="flang_3.html#E1:34"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="flang_2.html#T20" title="FLANG_2:th.20">FLANG_2:20</a></i>;<br/>
<a NAME="E11:34_2"/>
<font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="ref" href="flang_3.html#T3" target="_self" title="FLANG_3:th.3">Th3</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>hence </b><a NAME="E12:34_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="flang_3.html#E9:34_2"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="flang_3.html#E10:34_2"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="flang_1.html#D1" title="FLANG_1:def.1">FLANG_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="flang_3.html#E2:34"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
