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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="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>









<b>assume </b><a NAME="E1:95"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:95"/><i><font color="Green">E63</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>

<a NAME="E3:95"/><i><font color="Green">E64</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:95"/><i><font color="Green">E65</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:95"><i><font color="Green">E62</font></i></a>, <a class="txt" href="rfunct_3.html#E2:95"><i><font color="Green">E63</font></i></a>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<b>then </b><i><font color="Green">E66</font></i>:  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>2</sub></font>,<a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="rfunct_3.html#T66" target="_self">Th66</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="rfunct_3.html#T71" target="_self">Th71</a></i>
;<br/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:95"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E4:95"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_1.html#T50">XBOOLE_1:50</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>
<b>then </b> <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:95"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#T66" target="_self">Th66</a></i>
<br/>.= 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:95"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rfunct_3.html#T66" target="_self">Th66</a></i>
;<br/>
<b>hence </b><a NAME="E8:95"/>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E5:95"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>


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