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

<b>set </b><font color="Maroon">Y</font> =  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> ;<br/>
<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 ;<br/>

<b>assume </b><a NAME="E2:17"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> is  not  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> 
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:17"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:17"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  : <font color="Olive">x</font> is  <a href="realset2.html#V3">trivial</a>  <a href="orders_2.html#L1">RelStr</a>  </span>}</span> 
 <b>and </b><br/><a NAME="E6:17"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S3">RECDEF_1:sch 3</a>();<br/></i>
<a NAME="E7:17"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">x</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="E1:17_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:17_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:17_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; ( <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:17_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:17"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:17_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E5:17_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E6:17_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:17_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:17_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_2"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E8:17_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:17"/><i><font color="Green">E46</font></i>: 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:17_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:17_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> ;<br/>
<a NAME="E5:17_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:17_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0
 ;<br/>

<a NAME="E2:17_3_1"/><b>then </b>
 ex <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> is  <a href="realset2.html#V3">trivial</a>  <a href="orders_2.html#L1">RelStr</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:17_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:17_3"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_3_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:17_3_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:17_3_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:17_3_2"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon">x</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="E1:17_3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> <b> such that </b><br/><a NAME="E3:17_3_2_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) )
 ;<br/>
<b>thus </b><a NAME="E4:17_3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:17_3_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E8:17_3"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<a NAME="E9:17_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 
;<br/>
<b>hence </b><a NAME="E10:17_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> =  <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="neckla_2.html#K3" target="_self">fin_RelStr</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:17"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">R</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>   st not the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> is <a href="realset1.html#V1">trivial</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  holds <br/><font color="Olive">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">R</font> be  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_4"/><i><font color="Green">E85</font></i>: 
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> is <a href="realset1.html#V1">trivial</a> )
 <b>and </b><br/><a NAME="E2:17_4"/><i><font color="Green">E86</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a> 
 ;<br/>

<a NAME="E3:17_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">R</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:17_4"/>
<font color="Maroon">R</font> is  <a href="realset2.html#V3">trivial</a>  <a href="orders_2.html#L1">RelStr</a> 
 
<b>by </b><i>, <a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<a NAME="E5:17_4"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:17_4"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:17_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0
 ;<br/>

<b>hence </b><a NAME="E2:17_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:17_4"/>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:17"/>
for <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>   st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/>(  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp;  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H1</font> be  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">H2</font> be  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_5"/><i><font color="Green">E89</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>and </b><br/><a NAME="E2:17_5"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">H1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E3:17_5"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:17_5"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:17_5"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">H1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:17_5"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E9:17_5"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E11:17_5"/>
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> is   <a href="nat_1.html#NM1">Nat</a>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T1">FINSEQ_2:1</a></i>;<br/>
<a NAME="E12:17_5"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E13:17_5"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="axioms.html#T28">AXIOMS:28</a></i>;<br/>
<a NAME="E14:17_5"/>
<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E15:17_5"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="neckla_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E16:17_5"/>
<font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E17:17_5"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="neckla_2.html#T2" target="_self">Th2</a></i>;<br/>
<b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> ;<br/>
<a NAME="E18:17_5"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="neckla_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E19:17_5"/>
 <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E20:17_5"/><b>then </b>
 <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> 
 
<b>by </b><i>, , , , <a class="txt" href="neckla_2.html#E1:17"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E21:17_5"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E22:17_5"/>
 <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E23:17_5"/><b>then </b>
 <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> 
 
<b>by </b><i>, , , , <a class="txt" href="neckla_2.html#E1:17"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E24:17_5"/><b>then </b>
 <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E25:17_5"/>
(  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp;  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E2:17"><i><font color="Green">E32</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:17"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:17"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E15:17"/>
 ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:17_6"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:17_6"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> implies  ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> ) ) );<br/>
<a NAME="E4:17_6"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_6_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> <b> such that </b><br/><a NAME="E3:17_6_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:17_6_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">a</font> is  <a href="realset2.html#V3">trivial</a>  <a href="orders_2.html#L1">RelStr</a> 
 <b>by </b><i/>;<br/>
<a NAME="E5:17_6_1"/>
 ex <font color="Olive">R</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">R</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K13">FinSETS</a>  )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<b>thus </b><a NAME="E7:17_6_1"/>
 ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> ) )
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17_6"/><i><font color="Green">E122</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_6_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E2:17_6_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:17_6_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_6_2_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> or <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="neckla_2.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_6_2_1_1"><b>suppose </b></a><a NAME="E1:17_6_2_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_6_2_1_1"/>
 ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_6_2_1_2"><b>suppose </b></a><a NAME="E1:17_6_2_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  :  ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a> <b> such that </b><br/><a NAME="E3:17_6_2_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:17_6_2_1_2"/><i><font color="Green">E127</font></i>: 
 ex <font color="Olive">R1</font>, <font color="Olive">R2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R2</font> &amp; <font color="Olive">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">R1</font>,<font color="Olive">R2</font> ) )
 ;<br/><b>consider </b><font color="Maroon">R1</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">R2</font> being  <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E6:17_6_2_1_2"/><i><font color="Green">E128</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R2</font>
 <b>and </b><br/><a NAME="E7:17_6_2_1_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Maroon">R1</font>,<font color="Maroon">R2</font> or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Maroon">R1</font>,<font color="Maroon">R2</font> ) )
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E1:17"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E8:17_6_2_1_2"/>
( <font color="Maroon">R1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Maroon">R2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  )
 <b>by </b><i>, , <a class="txt" href="neckla_2.html#E2:17"><i><font color="Green">E32</font></i></a>, <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><b>hence </b><a NAME="E9:17_6_2_1_2"/>
 ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> ) )
 <b>by </b><i>, <a class="txt" href="neckla_2.html#E2:17"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:17_6"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">y</font> is   <a href="nat_1.html#NM1">Nat</a>
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:17_6"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E8:17_6"/>
 ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> ) )
 <b>by </b><i><a class="ref" href="neckla_2.html#T1" target="_self">Th1</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:17"/>
 ex <font color="Olive">H1</font>, <font color="Olive">H2</font> being <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H2</font> &amp; <font color="Olive">H1</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; <font color="Olive">H2</font> <a href="hidden.html#R2">in</a>  <a href="neckla_2.html#K4" target="_self">fin_RelStr_sp</a>  &amp; ( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K1" target="_self">union_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2" target="_self">sum_of</a> <font color="Olive">H1</font>,<font color="Olive">H2</font> ) )
 ;<br/>


</div>
