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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">CNS</font>,<font color="Maroon" title="c3">RNS</font>;<br/>









<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E89">A1</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<a NAME="E2:115"/><b>then </b><i><font color="Green" title="E90">A2</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="ncfcont1.html#T63" target="_self" title="NCFCONT1:th.63">Th63</a></i>;<br/>
<a NAME="E3:115"/><b>then </b><i><font color="Green" title="E91">A3</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#D4" title="VFUNCT_1:def.4">VFUNCT_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">s1</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">CNS</font>;<br/><b>assume </b><a NAME="E1:115_1"/><i><font color="Green" title="E92">A4</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp;  <a href="clvect_1.html#K8" title="CLVECT_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 ;<br/><a NAME="E2:115_1"/><b>then </b><i><font color="Green" title="E93">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ncfcont1.html#E2:115"><i><font color="Green" title="E90">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:115_1"/><i><font color="Green" title="E94">A6</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c5">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:115"><i><font color="Green" title="E89">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E1:115_1"><i><font color="Green" title="E92">A4</font></i></a>, <a class="ref" href="ncfcont1.html#T63" target="_self" title="NCFCONT1:th.63">Th63</a></i>;<br/><a NAME="E4:115_1"/><b>then </b><i><font color="Green" title="E95">A7</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#T37" title="NORMSP_1:th.37">NORMSP_1:37</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> = 
<font color="Maroon" title="c1">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:115"><i><font color="Green" title="E91">A3</font></i></a>, <a class="txt" href="ncfcont1.html#E1:115_1"><i><font color="Green" title="E92">A4</font></i></a>, <a class="txt" href="ncfcont1.html#E3:115_1"><i><font color="Green" title="E94">A6</font></i></a>, <a class="ref" href="vfunct_1.html#D4" title="VFUNCT_1:def.4">VFUNCT_1:def 4</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:115_1"><i><font color="Green" title="E94">A6</font></i></a>, <a class="ref" href="normsp_1.html#T45" title="NORMSP_1:th.45">NORMSP_1:45</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:115_1"><i><font color="Green" title="E93">A5</font></i></a>, <a class="ref" href="ncfcont1.html#T48" target="_self" title="NCFCONT1:th.48">Th48</a></i>
;<br/><b>hence </b><a NAME="E6:115_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K8" title="CLVECT_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="ncfcont1.html#K5" title="NCFCONT1:func.5">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E2:115_1"><i><font color="Green" title="E93">A5</font></i></a>, <a class="txt" href="ncfcont1.html#E4:115_1"><i><font color="Green" title="E95">A7</font></i></a>, <a class="ref" href="ncfcont1.html#T48" target="_self" title="NCFCONT1:th.48">Th48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:115"/>
<font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c5">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:115"><i><font color="Green" title="E91">A3</font></i></a>, <a class="ref" href="ncfcont1.html#T63" target="_self" title="NCFCONT1:th.63">Th63</a></i>;<br/>


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