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<b>let </b><font color="Maroon" title="c1">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">S</font>, <font color="Olive" title="b2">T</font> being  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</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="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font>  st <font color="Olive" title="b3">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font> holds <br/>( <span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Olive" title="b3">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="nfcont_1.html#R4" title="NFCONT_1:pred.4">is_continuous_on</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Olive" title="b3">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font> )</span><br/><b>let </b><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">T</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">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b1">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font> holds <br/>( <span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="nfcont_1.html#R4" title="NFCONT_1:pred.4">is_continuous_on</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Olive" title="b1">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font> )</span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font> implies ( <span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="nfcont_1.html#R4" title="NFCONT_1:pred.4">is_continuous_on</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font> ) )</span><br/>







<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="nfcont_1.html#R4" title="NFCONT_1:pred.4">is_continuous_on</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>thus </b><a NAME="E2:46"/>
<span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="nfcont_1.html#R4" title="NFCONT_1:pred.4">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_1"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c1">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="c4">f</font>
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:46"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="nfcont_1.html#D11" target="_self" title="NFCONT_1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E2:46_1"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c1">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"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D2" target="_self" title="NFCONT_1:def.2">Def2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D12" target="_self" title="NFCONT_1:def.12">NFCONT_1:def 12</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">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="nfcont_1.html#R2" title="NFCONT_1:pred.2">is_continuous_in</a> <font color="Olive" title="b1">x0</font></span><br/>

<b>let </b><font color="Maroon" title="c5">r</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</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">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="nfcont_1.html#R2" title="NFCONT_1:pred.2">is_continuous_in</a> <font color="Maroon" title="c5">r</font> )</span><br/>

<b>assume </b><a NAME="E3:46_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="nfcont_1.html#R2" title="NFCONT_1:pred.2">is_continuous_in</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E4:46_1"/><b>then </b><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:46"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="nfcont_1.html#D11" target="_self" title="NFCONT_1:def.11">Def11</a></i>;<br/>
<b>thus </b><a NAME="E5:46_1"/>
<span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="nfcont_1.html#R2" title="NFCONT_1:pred.2">is_continuous_in</a> <font color="Maroon" title="c5">r</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:46_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="nfcont_1.html#E3:46_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:46_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D10" target="_self" title="NFCONT_1:def.10">NFCONT_1:def 10</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">s1</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Olive" title="b1">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="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Olive" title="b1">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Olive" title="b1">s1</font></span>)</span> )</span><br/>

<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">S</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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> &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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> implies ( <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E3:46_1_1"/><i><font color="Green" title="E43">A7</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> <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> &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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</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"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )</span><br/>

<a NAME="E4:46_1_1"/><b>then </b>
 <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> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E5:46_1_1"/><b>then </b>
 <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> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#D2" target="_self" title="NFCONT_1:def.2">Def2</a></i>;<br/>
<a NAME="E6:46_1_1"/><b>then </b><i><font color="Green" title="E44">A8</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E7:46_1_1"/><b>then </b><i><font color="Green" title="E45">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</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="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font> <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="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E4:46_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="nfcont_1.html#E3:46_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self" title="NFCONT_1:def.9">Def9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span><br/><a NAME="E1:46_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E2:46_1_1_1"/><b>then </b><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:46_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E6:46_1_1"><i><font color="Green" title="E44">A8</font></i></a></i>;<br/><a NAME="E4:46_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E5:46_1_1_1"/><b>then </b><i><font color="Green" title="E47">A11</font></i>: 
( <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:46_1_1_1"/><b>then </b><i><font color="Green" title="E48">A12</font></i>: 
( <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> &amp; <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D2" target="_self" title="NFCONT_1:def.2">Def2</a></i>;<br/><b>thus </b><span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E6:46_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="nfcont_1.html#T8" target="_self" title="NFCONT_1:th.8">Th8</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E6:46_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="nfcont_1.html#E2:46_1_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E6:46_1_1_1"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="nfcont_1.html#D2" target="_self" title="NFCONT_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E5:46_1_1_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E3:46_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="nfcont_1.html#E2:46_1_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E3:46_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="nfcont_1.html#T9" target="_self" title="NFCONT_1:th.9">Th9</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>
<a NAME="E9:46_1_1"/><b>then </b><i><font color="Green" title="E46">A13</font></i>: 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E10:46_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E7:46_1_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="normsp_1.html#T39" title="NORMSP_1:th.39">NORMSP_1:39</a></i>; <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"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="nfcont_1.html#K3" title="NFCONT_1:func.3">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span><br/>

<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:46_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="nfcont_1.html#E3:46_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="partfun2.html#T35" title="PARTFUN2:th.35">PARTFUN2:35</a></i>
<br/>.= 
<span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:46_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="nfcont_1.html#E3:46_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="nfcont_1.html#D2" target="_self" title="NFCONT_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:46_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="nfcont_1.html#E3:46_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:46_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="partfun2.html#T34" title="PARTFUN2:th.34">PARTFUN2:34</a></i>
;<br/>
<b>hence </b><a NAME="E12:46_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</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#E7:46_1_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="nfcont_1.html#E9:46_1_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="lopban_1.html#T24" title="LOPBAN_1:th.24">LOPBAN_1:24</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>


</div><b>end;</b></div>

<a NAME="E3:46"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:46"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="nfcont_1.html#T34" target="_self" title="NFCONT_1:th.34">Th34</a></i>;<br/>
<b>hence </b><a NAME="E4:46"/>
 <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="vfunct_1.html#T29" title="VFUNCT_1:th.29">VFUNCT_1:29</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>
