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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green">E17</font></i>: 
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><a NAME="E2:28_1_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="afinsq_1.html#K2">dom</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green">E17</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/><a NAME="E4:28_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green">E17</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="stirl2_1.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E5:28_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:28_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green">E17</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green">E17</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E5:28_1_1_2"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<a NAME="E1:28_1_1_2_1"/>
(  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E2:28_1_1_2_1"/><b>then </b>
(  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="ordinal1.html#V7">natural</a> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E3:28_1_1_2_1"/><b>then </b>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E7:28_1_1_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="stirl2_1.html#E5:28_1_1_2"><i><font color="Green">E19</font></i></a>);<br/></i><a NAME="E8:28_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_2.html#V2">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_2_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:28_1_1_2_2_1"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="stirl2_1.html#T8" target="_self">Th8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_2_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E4:28_1_1_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E5:28_1_1_2_2_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:28_1_1_2_2_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="square_1.html#K1">min</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><a NAME="E7:28_1_1_2_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E8:28_1_1_2_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:28_1_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:28_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:28_1_1_2"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> holds <br/> <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:28_1_1_2_3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:28_1_1_2_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E3:28_1_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:28"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:28_1_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="square_1.html#K1">min</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/>
<a NAME="E5:28_1_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:28_1_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:28_1_1_2_3"/><b>then </b><i><font color="Green">E25</font></i>: 
( not <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E8:28_1_1_2_3"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:28_1_1_2_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_2_3_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="stirl2_1.html#E8:28_1_1_2_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1_1_2_3_1_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:28_1_1_2_3_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:28_1_1_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T8" target="_self">Th8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E5:28_1_1_2_3_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E6:28_1_1_2_3_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:28_1_1_2_3_1_1_1_1"/><b>then </b>
 <a href="square_1.html#K1">min</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E8:28_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_1"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E9:28_1_1_2_3_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E3:28_1_1_2_3_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_2_3_1_1_1"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2_3_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_2_3_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:28_1_1_2_3_1_1_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:28_1_1_2_3_1_1_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:28_1_1_2_3_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_2_3_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_1_2_3_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_1_2_3_1_1_2"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E7:28_1_1_2_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:28_1_1_2_3"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:28_1_1_2_4"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:28_1_1_2_4"/>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E9:28_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="stirl2_1.html#E1:28_1_1_2_4"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_2_4"/>
 <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="stirl2_1.html#K1" target="_self">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:28_1_1_2_4"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E11:28_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a>
 <b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E12:28_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E8:28_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="stirl2_1.html#V1" target="_self">"increasing</a> )
 ;<br/>


</div>
