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<a NAME="E1:59_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c4">Key</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">r</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c4">Key</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:59_1_1_1"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c4">Key</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">r</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">xx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:59_1_1_1"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c6">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ;<br/>
<a NAME="E4:59_1_1_1"/>
<span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c4">Key</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">Key</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c6">xx</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:59_1_1_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="idea_1.html#E3:59_1_1_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E5:59_1_1_1"/>
<span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c4">Key</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:59_1_1_1"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:59_1_1"/>
 <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c4">Key</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">r</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 <b>by </b><i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></i>;<br/>


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