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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_1.html#V2">non-empty</a> non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="hallmar1.html#M4" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="hallmar1.html#M4" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E3:40_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="hallmar1.html#M2" target="_self">Reduction</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>hence </b><a NAME="E4:40_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="hallmar1.html#D6" target="_self">Def6</a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E5:40_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E6:40_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="hallmar1.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E7:40_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="hallmar1.html#E5:40_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="hallmar1.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E8:40_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E5:40_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="hallmar1.html#E6:40_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hallmar1.html#D7" target="_self">Def7</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:40"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<a NAME="E3:40"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="hallmar1.html#E2:40"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E5:40"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:40_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:40_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E2:40"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:40_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:40_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hallmar1.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="hallmar1.html#M2" target="_self">Reduction</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="hallmar1.html#E2:40"><i><font color="Green">E33</font></i></a>, <a class="ref" href="hallmar1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E7:40"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:40_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:40_3"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="hallmar1.html#M3" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E2:40"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E3:40_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:40_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="hallmar1.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E4:40_3"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:40_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="hallmar1.html#E2:40_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="hallmar1.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:40"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="hallmar1.html#M4" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="hallmar1.html#D8" target="_self">Def8</a></i>;<br/>


</div>
