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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:5"/><i><font color="Green">E6</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:5"/><i><font color="Green">E7</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 </span>}</span> 
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 
;<br/>
<b>hence </b><a NAME="E3:5_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:5"/>
2 <a href="nat_1.html#K2">*</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="scmpds_9.html#E1:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="scmpds_9.html#E2:5"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E5:5"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:5"/><i><font color="Green">E8</font></i>: 
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E8:5"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>
<a NAME="E9:5"/><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E7:5"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E10:5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E11:5"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E8:5"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E12:5"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E9:5"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E13:5"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:5"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>


</div>
