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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">CNS</font> being  <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a><br/> for <font color="Olive" title="b2">RNS</font> being  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/> for <font color="Olive" title="b3">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Olive" title="b1">CNS</font>,<font color="Olive" title="b2">RNS</font>  st <font color="Olive" title="b4">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b3">X</font> holds <br/><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b4">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b3">X</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">RNS</font> being  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/> for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">CNS</font>,<font color="Olive" title="b1">RNS</font>  st <font color="Olive" title="b3">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b2">X</font> holds <br/><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b3">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b2">X</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <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>  st <font color="Olive" title="b2">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b2">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b1">X</font></span><br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <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>  st <font color="Olive" title="b1">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Maroon" title="c1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b1">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">X</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies <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> )</span><br/>









<b>assume </b><a NAME="E1:88"/><i><font color="Green" title="E64">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></span><br/>

<a NAME="E2:88"/><b>then </b><i><font color="Green" title="E65">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="E3:88"/><b>then </b><i><font color="Green" title="E66">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="88_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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#K9" title="CLVECT_1:func.9">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> implies ( <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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> ) )</span><br/><b>assume </b><a NAME="E1:88_1"/><i><font color="Green" title="E67">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#K9" title="CLVECT_1:func.9">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )</span><br/><a NAME="E2:88_1"/><b>then </b><i><font color="Green" title="E68">A5</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> &amp; <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:88"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="ncfcont1.html#T63" target="_self" title="NCFCONT1:th.63">Th63</a></i>;<br/><a NAME="E3:88_1"/><b>then </b><i><font color="Green" title="E69">A6</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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">lim</a> <font color="Maroon" title="c6">s1</font></span>)</span> = 
<font color="Maroon" title="c1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:88"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="#E1:88_1"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="#E2:88_1"><i><font color="Green" title="E68">A5</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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:88_1"><i><font color="Green" title="E68">A5</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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:88"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="#E1:88_1"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="ncfcont1.html#T48" target="_self" title="NCFCONT1:th.48">Th48</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>
;<br/><b>hence </b><a NAME="E5:88_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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K9" title="CLVECT_1:func.9">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="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:88"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="#E1:88_1"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="#E3:88_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="ncfcont1.html#T48" target="_self" title="NCFCONT1:th.48">Th48</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:88"/>
<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="#E3:88"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="ncfcont1.html#T63" target="_self" title="NCFCONT1:th.63">Th63</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
