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<a NAME="E1:58_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#K17" title="IDEA_1:func.17">IDEA_P_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#K17" title="IDEA_1:func.17">IDEA_P_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="58_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K17" title="IDEA_1:func.17">IDEA_P_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> implies <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_1"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<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#K17" title="IDEA_1:func.17">IDEA_P_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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:58_1_1_1"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<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="E4:58_1_1_1"/>
<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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#K15" title="IDEA_1:func.15">IDEA_P</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>,<font color="Maroon" title="c6">xx</font></span>)</span>,<font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E3:58_1_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:58_1_1_1"/>
<span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:58_1_1"/>
 <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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