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<a NAME="E1:57_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> is   <a href="funct_1.html#NM1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:57_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:57_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E4:57_1_1_1"/>
<span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K15" target="_self">IDEA_P</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">Key</font>,<font color="Maroon">xx</font></span>)</span>,<font color="Maroon">n</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:57_1_1_1"/>
<span class="p1">(<span class="default"><a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_1_1"/>
 <a href="idea_1.html#K17" target="_self">IDEA_P_F</a> <font color="Maroon">Key</font>,<font color="Maroon">n</font>,<font color="Maroon">r</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>


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