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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="nfcont_1.html#NM1" title="NFCONT_1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>;<br/>









<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<a NAME="E2:47"/><b>then </b><i><font color="Green" title="E38">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="nfcont_1.html#T25" target="_self" title="NFCONT_1:th.25">Th25</a></i>;<br/>
<a NAME="E3:47"/><b>then </b><i><font color="Green" title="E39">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="c3">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="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">s1</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c2">S</font>;<br/><b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E40">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="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_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="E2:47_1"/><b>then </b><i><font color="Green" title="E41">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="nfcont_1.html#E2:47"><i><font color="Green" title="E38">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="E3:47_1"/><i><font color="Green" title="E42">A6</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="nfcont_1.html#K3" title="NFCONT_1:func.3">*</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="normsp_1.html#K8" title="NORMSP_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="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:47"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="nfcont_1.html#E1:47_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="nfcont_1.html#T25" target="_self" title="NFCONT_1:th.25">Th25</a></i>;<br/><a NAME="E4:47_1"/><b>then </b><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c3">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="nfcont_1.html#K3" title="NFCONT_1:func.3">*</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="c3">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="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> = 
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</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="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E3:47"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="nfcont_1.html#E1:47_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="nfcont_1.html#E3:47_1"><i><font color="Green" title="E42">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="c3">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="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E3:47_1"><i><font color="Green" title="E42">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="c3">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="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:47_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="nfcont_1.html#T20" target="_self" title="NFCONT_1:th.20">Th20</a></i>
;<br/><b>hence </b><a NAME="E6:47_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="nfcont_1.html#K3" title="NFCONT_1:func.3">*</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="c3">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="normsp_1.html#K8" title="NORMSP_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="c3">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="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E2:47_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="nfcont_1.html#E4:47_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="nfcont_1.html#T20" target="_self" title="NFCONT_1:th.20">Th20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
<font color="Maroon" title="c3">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c5">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E3:47"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="nfcont_1.html#T25" target="_self" title="NFCONT_1:th.25">Th25</a></i>;<br/>


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