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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">H</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">H</font> <a href="seqfunc.html#R3" title="SEQFUNC:pred.3">is_unif_conv_on</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E2:39"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="seqfunc.html#D13" target="_self" title="SEQFUNC:def.13">Def13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:39_1"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> ) )
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:39"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="seqfunc.html#D13" target="_self" title="SEQFUNC:def.13">Def13</a></i>;<br/><b>take </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font>;<br/><b>thus </b><a NAME="E3:39_1"/>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:39_1"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>assume </b><a NAME="E4:39_1"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><b>let </b><font color="Maroon" title="c7">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E5:39_1"/>
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c8">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:39_1"/><i><font color="Green" title="E31">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:39_1"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">k</font>;<br/><b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E8:39_1"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font>
 ;<br/><b>hence </b><a NAME="E9:39_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E4:39_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="seqfunc.html#E7:39_1"><i><font color="Green" title="E31">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
<font color="Maroon" title="c2">H</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:39"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="seqfunc.html#D12" target="_self" title="SEQFUNC:def.12">Def12</a></i>;<br/>


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