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

<b>set </b><font color="Maroon" title="c1">Y</font> =  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> ;<br/>
<b>let </b><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 ;<br/>

<b>assume </b><a NAME="E2:17"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c2">X</font> is  not  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> 
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $2 <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:17"/><i><font color="Green" title="E13">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E5:17"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  : <font color="Olive" title="b1">x</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  </span>}</span> 
 <b>and </b><br/><a NAME="E6:17"/><i><font color="Green" title="E15">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S3" title="RECDEF_1:sch.3">RECDEF_1:sch 3</a>();<br/></i>
<a NAME="E7:17"/><i><font color="Green" title="E16">A6</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>



<b>assume </b><a NAME="E1:17_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 ;<br/>

<a NAME="E2:17_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:17_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; ( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E6:17"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:17_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:17_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:17"/><i><font color="Green" title="E17">A8</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">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" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_2"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:17_2"/><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c6">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span>;<br/>
<a NAME="E5:17_2"/><i><font color="Green" title="E19">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E6:17_2"/><i><font color="Green" title="E20">A11</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_2_1"/><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]
 ;<br/>

<a NAME="E2:17_2_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E7:17"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:17_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:17_2_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17_2"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="neckla_2.html#E5:17_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="neckla_2.html#E6:17_2"><i><font color="Green" title="E20">A11</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:17_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:17_2"><i><font color="Green" title="E18">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:17"/><i><font color="Green" title="E18">A13</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">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" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_3"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_3"/><i><font color="Green" title="E19">A14</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E4:17_3"/><i><font color="Green" title="E20">A15</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> ;<br/>
<a NAME="E5:17_3"/><i><font color="Green" title="E21">A16</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" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_3_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
 ;<br/>

<a NAME="E2:17_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  st <br/>( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  )
 
<b>by </b><i><a class="txt" href="neckla_2.html#E5:17"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:17_3_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:17_3"/><i><font color="Green" title="E22">A17</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_3_2"/><i><font color="Green" title="E23">A18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ;<br/>

<a NAME="E2:17_3_2"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:17_3_2"/><b>then </b><i><font color="Green" title="E24">A19</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E6:17"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>
<a NAME="E4:17_3_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">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" title="c7">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_3_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> <b> such that </b><br/><a NAME="E3:17_3_2_1"/><i><font color="Green" title="E25">A20</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp;  ex <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R2</font> &amp; <font color="Olive" title="b1">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b2">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">R1</font>,<font color="Olive" title="b2">R2</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">R1</font>,<font color="Olive" title="b2">R2</font> ) ) )
 ;<br/>
<b>thus </b><a NAME="E4:17_3_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:17_3_2_1"><i><font color="Green" title="E25">A20</font></i></a></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" title="c6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:17_3_2"><i><font color="Green" title="E23">A18</font></i></a>, <a class="txt" href="neckla_2.html#E3:17_3_2"><i><font color="Green" title="E24">A19</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="neckla_2.html#E4:17"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="neckla_2.html#E3:17_3"><i><font color="Green" title="E19">A14</font></i></a></i>;<br/>
<a NAME="E8:17_3"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="neckla_2.html#E5:17_3"><i><font color="Green" title="E21">A16</font></i></a>, <a class="txt" href="neckla_2.html#E6:17_3"><i><font color="Green" title="E22">A17</font></i></a>);<br/></i>
<a NAME="E9:17_3"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 
;<br/>
<b>hence </b><a NAME="E10:17_3"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 <b>by </b><i><a class="txt" href="neckla_2.html#E4:17_3"><i><font color="Green" title="E20">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">M</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">f</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:17"/><i><font color="Green" title="E19">A21</font></i>: 
for <font color="Olive" title="b1">R</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  holds <br/><font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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" title="c5">R</font> be  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_4"/><i><font color="Green" title="E20">A22</font></i>: 
( not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">R</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>and </b><br/><a NAME="E2:17_4"/><i><font color="Green" title="E21">A23</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 ;<br/>

<a NAME="E3:17_4"/><i><font color="Green" title="E22">A24</font></i>: 
<font color="Maroon" title="c5">R</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E1:17_4"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="neckla_2.html#E2:17_4"><i><font color="Green" title="E21">A23</font></i></a>, <a class="ref" href="neckla_2.html#D5" target="_self" title="NECKLA_2:def.5">Def5</a></i>;<br/>
<a NAME="E4:17_4"/>
<font color="Maroon" title="c5">R</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E1:17_4"><i><font color="Green" title="E20">A22</font></i></a>, <a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/>
<a NAME="E5:17_4"/><b>then </b><i><font color="Green" title="E23">A25</font></i>: 
<font color="Maroon" title="c5">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
 
<b>by </b><i><a class="txt" href="neckla_2.html#E5:17"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="neckla_2.html#E3:17_4"><i><font color="Green" title="E22">A24</font></i></a></i>;<br/>
<a NAME="E6:17_4"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_4_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
 ;<br/>

<b>hence </b><a NAME="E2:17_4_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E4:17"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:17_4"/>
<font color="Maroon" title="c5">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E5:17_4"><i><font color="Green" title="E23">A25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:17"/>
for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> holds <br/>(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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" title="c5">H1</font>, <font color="Maroon" title="c6">H2</font> be  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_5"/><i><font color="Green" title="E20">A26</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H2</font>
 <b>and </b><br/><a NAME="E2:17_5"/><i><font color="Green" title="E21">A27</font></i>: 
<font color="Maroon" title="c5">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <b>and </b><br/><a NAME="E3:17_5"/><i><font color="Green" title="E22">A28</font></i>: 
<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:17_5"/><i><font color="Green" title="E23">A29</font></i>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E6:17_5"/><i><font color="Green" title="E24">A30</font></i>: 
<font color="Maroon" title="c5">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E2:17_5"><i><font color="Green" title="E21">A27</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:17_5"/><i><font color="Green" title="E25">A31</font></i>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E9:17_5"/><i><font color="Green" title="E26">A32</font></i>: 
<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:17_5"><i><font color="Green" title="E22">A28</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c10">x2</font> = <font color="Maroon" title="c8">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="neckla_2.html#E4:17"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="neckla_2.html#E5:17_5"><i><font color="Green" title="E23">A29</font></i></a>, <a class="txt" href="neckla_2.html#E8:17_5"><i><font color="Green" title="E25">A31</font></i></a></i>;<br/>
<a NAME="E11:17_5"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T1" title="FINSEQ_2:th.1">FINSEQ_2:1</a></i>;<br/>
<a NAME="E12:17_5"/><b>then </b><i><font color="Green" title="E27">A33</font></i>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E13:17_5"/><b>then </b><i><font color="Green" title="E28">A34</font></i>: 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E4:17"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="axioms.html#T28" title="AXIOMS:th.28">AXIOMS:28</a></i>;<br/>
<a NAME="E14:17_5"/>
<font color="Maroon" title="c9">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E15:17_5"/><b>then </b><i><font color="Green" title="E29">A35</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E8:17"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="neckla_2.html#E12:17_5"><i><font color="Green" title="E27">A33</font></i></a></i>;<br/>
<a NAME="E16:17_5"/>
<font color="Maroon" title="c10">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E17:17_5"/><b>then </b><i><font color="Green" title="E30">A36</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E8:17"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="neckla_2.html#E12:17_5"><i><font color="Green" title="E27">A33</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">W</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> ;<br/>
<a NAME="E18:17_5"/><i><font color="Green" title="E31">A37</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E6:17"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="neckla_2.html#E12:17_5"><i><font color="Green" title="E27">A33</font></i></a></i>;<br/>
<a NAME="E19:17_5"/>
 <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E9:17"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="neckla_2.html#E1:17_5"><i><font color="Green" title="E20">A26</font></i></a>, <a class="txt" href="neckla_2.html#E2:17_5"><i><font color="Green" title="E21">A27</font></i></a>, <a class="txt" href="neckla_2.html#E3:17_5"><i><font color="Green" title="E22">A28</font></i></a>, <a class="ref" href="neckla_2.html#D5" target="_self" title="NECKLA_2:def.5">Def5</a></i>;<br/>
<a NAME="E20:17_5"/><b>then </b>
 <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E1:17_5"><i><font color="Green" title="E20">A26</font></i></a>, <a class="txt" href="neckla_2.html#E6:17_5"><i><font color="Green" title="E24">A30</font></i></a>, <a class="txt" href="neckla_2.html#E9:17_5"><i><font color="Green" title="E26">A32</font></i></a>, <a class="txt" href="neckla_2.html#E15:17_5"><i><font color="Green" title="E29">A35</font></i></a>, <a class="txt" href="neckla_2.html#E17:17_5"><i><font color="Green" title="E30">A36</font></i></a></i>;<br/>
<a NAME="E21:17_5"/><b>then </b><i><font color="Green" title="E32">A38</font></i>: 
 <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E18:17_5"><i><font color="Green" title="E31">A37</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E22:17_5"/>
 <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E9:17"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="neckla_2.html#E1:17_5"><i><font color="Green" title="E20">A26</font></i></a>, <a class="txt" href="neckla_2.html#E2:17_5"><i><font color="Green" title="E21">A27</font></i></a>, <a class="txt" href="neckla_2.html#E3:17_5"><i><font color="Green" title="E22">A28</font></i></a>, <a class="ref" href="neckla_2.html#D5" target="_self" title="NECKLA_2:def.5">Def5</a></i>;<br/>
<a NAME="E23:17_5"/><b>then </b>
 <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E1:17_5"><i><font color="Green" title="E20">A26</font></i></a>, <a class="txt" href="neckla_2.html#E6:17_5"><i><font color="Green" title="E24">A30</font></i></a>, <a class="txt" href="neckla_2.html#E9:17_5"><i><font color="Green" title="E26">A32</font></i></a>, <a class="txt" href="neckla_2.html#E15:17_5"><i><font color="Green" title="E29">A35</font></i></a>, <a class="txt" href="neckla_2.html#E17:17_5"><i><font color="Green" title="E30">A36</font></i></a></i>;<br/>
<a NAME="E24:17_5"/><b>then </b>
 <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E18:17_5"><i><font color="Green" title="E31">A37</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E25:17_5"/>
(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E13:17_5"><i><font color="Green" title="E28">A34</font></i></a>, <a class="txt" href="neckla_2.html#E21:17_5"><i><font color="Green" title="E32">A38</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:17"/><b>then </b><i><font color="Green" title="E20">A39</font></i>: 
 <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">M</font>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E11:17"><i><font color="Green" title="E19">A21</font></i></a>, <a class="ref" href="neckla_2.html#D5" target="_self" title="NECKLA_2:def.5">Def5</a></i>;<br/>
<a NAME="E14:17"/><b>then </b><i><font color="Green" title="E21">A40</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E9:17"><i><font color="Green" title="E18">A13</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E15:17"/>
 ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">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" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:17_6"/><i><font color="Green" title="E22">A41</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E3:17_6"/><i><font color="Green" title="E23">A42</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:17"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="neckla_2.html#E13:17"><i><font color="Green" title="E20">A39</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 implies  ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> ) ) );<br/>
<a NAME="E4:17_6"/><i><font color="Green" title="E24">A43</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" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> <b> such that </b><br/><a NAME="E3:17_6_1"/><i><font color="Green" title="E25">A44</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>and </b><br/><a NAME="E4:17_6_1"/><i><font color="Green" title="E26">A45</font></i>: 
<font color="Maroon" title="c6">a</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> 
 <b>by </b><i><a class="txt" href="neckla_2.html#E5:17"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/>
<a NAME="E5:17_6_1"/>
 ex <font color="Olive" title="b1">R</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">R</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  )
 
<b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self" title="NECKLA_2:def.4">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a</font> as    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<b>thus </b><a NAME="E7:17_6_1"/>
 ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> ) )
 <b>by </b><i><a class="txt" href="neckla_2.html#E2:17"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="neckla_2.html#E3:17_6_1"><i><font color="Green" title="E25">A44</font></i></a>, <a class="txt" href="neckla_2.html#E4:17_6_1"><i><font color="Green" title="E26">A45</font></i></a>, <a class="ref" href="neckla_2.html#T4" target="_self" title="NECKLA_2:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17_6"/><i><font color="Green" title="E25">A46</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_6_2"/><i><font color="Green" title="E26">A47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c7">W</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E2:17_6_2"/><i><font color="Green" title="E27">A48</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c6">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:17_6_2"/><i><font color="Green" title="E28">A49</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="neckla_2.html#E6:17"><i><font color="Green" title="E15">A5</font></i></a></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" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="neckla_2.html#E4:17_6_2"><i><font color="Green" title="E28">A49</font></i></a>, <a class="txt" href="neckla_2.html#E2:17_6_2"><i><font color="Green" title="E27">A48</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_6_2_1_1"/>
 ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> ) )
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:17_6_2"><i><font color="Green" title="E26">A47</font></i></a></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" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  :  ex <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b2">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b3">R2</font> ) ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> <b> such that </b><br/><a NAME="E3:17_6_2_1_2"/><i><font color="Green" title="E29">A50</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E4:17_6_2_1_2"/><i><font color="Green" title="E30">A51</font></i>: 
 ex <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R2</font> &amp; <font color="Olive" title="b1">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Olive" title="b2">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">R1</font>,<font color="Olive" title="b2">R2</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">R1</font>,<font color="Olive" title="b2">R2</font> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c10">R1</font>, <font color="Maroon" title="c11">R2</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E6:17_6_2_1_2"/><i><font color="Green" title="E31">A52</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R2</font>
 <b>and </b><br/><a NAME="E7:17_6_2_1_2"/><i><font color="Green" title="E32">A53</font></i>: 
( <font color="Maroon" title="c10">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c11">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c10">R1</font>,<font color="Maroon" title="c11">R2</font> or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c10">R1</font>,<font color="Maroon" title="c11">R2</font> ) )
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:17_6_2_1_2"><i><font color="Green" title="E29">A50</font></i></a>, <a class="txt" href="neckla_2.html#E4:17_6_2_1_2"><i><font color="Green" title="E30">A51</font></i></a></i>;<br/><a NAME="E8:17_6_2_1_2"/>
( <font color="Maroon" title="c10">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Maroon" title="c11">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  )
 <b>by </b><i><a class="txt" href="neckla_2.html#E4:17"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="neckla_2.html#E14:17"><i><font color="Green" title="E21">A40</font></i></a>, <a class="txt" href="neckla_2.html#E7:17_6_2_1_2"><i><font color="Green" title="E32">A53</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/><b>hence </b><a NAME="E9:17_6_2_1_2"/>
 ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> ) )
 <b>by </b><i><a class="txt" href="neckla_2.html#E6:17_6_2_1_2"><i><font color="Green" title="E31">A52</font></i></a>, <a class="txt" href="neckla_2.html#E7:17_6_2_1_2"><i><font color="Green" title="E32">A53</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" title="E26">A54</font></i>: 
<font color="Maroon" title="c5">y</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E4:17"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="neckla_2.html#E2:17_6"><i><font color="Green" title="E22">A41</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:17_6"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="neckla_2.html#E4:17_6"><i><font color="Green" title="E24">A43</font></i></a>, <a class="txt" href="neckla_2.html#E5:17_6"><i><font color="Green" title="E25">A46</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:17_6"/>
 ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> ) )
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:17_6"><i><font color="Green" title="E23">A42</font></i></a>, <a class="txt" href="neckla_2.html#E6:17_6"><i><font color="Green" title="E26">A54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:17"/>
 ex <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; ( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> ) )
 ;<br/>


</div>
