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

<a NAME="E1:39_1_1"/><i><font color="Green">E26</font></i>: 
0 <a href="hidden.html#R2">in</a> 1
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span></span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">1,1</span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">1,1</span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:39_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mcart_1.html#K3">[</a><span class="default">0,0,0</span><a href="mcart_1.html#K3">]</a></span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span>,<span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="mcart_1.html#K3">[</a><span class="default">0,0,0</span><a href="mcart_1.html#K3">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:39_1_1"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="mcart_1.html#K3">[</a><span class="default">0,0,0</span><a href="mcart_1.html#K3">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">1,1,1</span><a href="zfmisc_1.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="mcart_1.html#T39">MCART_1:39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">0,0,0</span><a href="mcart_1.html#K3">]</a></span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">1,1,1</span><a href="zfmisc_1.html#K3">:]</a></span> <b>by </b><i><a class="txt" href="altcat_1.html#E4:39_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:39_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:39_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:39_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">0,0,0</span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E4:39_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">1,1,1</span><a href="zfmisc_1.html#K3">:]</a></span> ;<br/>
<i><font color="Green">E29</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> 0,0 = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span>

<br/>.= 
 <a href="funct_2.html#K1">Funcs</a> 0,0
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">1,1,1</span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/><a NAME="E2:39_1_1_4"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default">0,0,0</span><a href="mcart_1.html#K3">]</a></span>
 <b>by </b><i><a class="txt" href="altcat_1.html#E5:39_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:39_1_1_4"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><i><font color="Green">E32</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> 0,0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> 0,0</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="altcat_1.html#E9:39_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_1.html#E3:39_1_1_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K3" target="_self">{|</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="altcat_1.html#K3" target="_self">|}</a></span> <a href="multop_1.html#K1">.</a> 0,0,0
<b>by </b><i><a class="txt" href="altcat_1.html#E1:39_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="altcat_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K3" target="_self">{|</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="altcat_1.html#K3" target="_self">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:39_1_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
;<br/><i><font color="Green">E33</font></i>: <span class="p1"><a href="altcat_1.html#K2" target="_self">{|</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="altcat_1.html#K2" target="_self">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1"><a href="altcat_1.html#K2" target="_self">{|</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="altcat_1.html#K2" target="_self">|}</a></span> <a href="multop_1.html#K1">.</a> 0,0,0
<b>by </b><i><a class="txt" href="altcat_1.html#E2:39_1_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> 0,0
<b>by </b><i><a class="txt" href="altcat_1.html#E1:39_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="altcat_1.html#D5" target="_self">Def5</a></i>
;<br/><a NAME="E7:39_1_1_4"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K2" target="_self">{|</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="altcat_1.html#K2" target="_self">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E9:39_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_1.html#E3:39_1_1_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="altcat_1.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E8:39_1_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3" target="_self">{|</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="altcat_1.html#K3" target="_self">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1"><a href="altcat_1.html#K2" target="_self">{|</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="altcat_1.html#K2" target="_self">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E9:39_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_1.html#E5:39_1_1_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_1.html#E6:39_1_1_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="altcat_1.html#NM3">BinComp</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>5</sub></font> =  <a href="altcat_1.html#G2" target="_self">AltCatStr</a>(# 1,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> #);<br/>

<b>thus </b><a NAME="E12:39_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="altcat_1.html#V6" target="_self">strict</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>5</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D15" target="_self">ALTCAT_1:def 15</a><br/>

<a NAME="E13:39_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:39_1_1"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> 0,0
 
<b>by </b><i><a class="txt" href="altcat_1.html#E9:39_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E15:39_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E13:39_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b>the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">c<sub>5</sub></font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 0,0</span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E13:39_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="altcat_1.html#E13:39_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="altcat_1.html#E15:39_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/>


</div>
