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<b>let </b><font color="Maroon">D</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">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<a NAME="E1:97"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="rfunct_3.html#D3" target="_self">Def3</a>);<br/></i>
<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:97"/><i><font color="Green">E31</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Olive">d1</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">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> &amp; <font color="Olive">d2</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">F</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font></span>)</span></span>)</span> holds <br/><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d2</font> ) )
 ;<br/>

<a NAME="E3:97"/><b>then </b>
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T104">RELAT_1:104</a></i>;<br/>
<a NAME="E4:97"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">dFY</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E6:97"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">dFY</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">dFY</font>
 
;<br/>
<b>hence </b><a NAME="E7:97"/>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</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">F</font>,<font color="Maroon">Y</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">F</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


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