<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">CNS</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">RNS</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">RNS</font>,<font color="Maroon">CNS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:123"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a>
 <b>and </b><br/><a NAME="E2:123"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font> holds  <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x2</font></span>)</span>
 ;<br/>

<b>given </b><font color="Maroon">x0</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font><b> such that </b><a NAME="E3:123"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="ncfcont1.html#R3" target="_self">is_continuous_in</a> <font color="Maroon">x0</font>
 ;<br/>

<a NAME="E4:123"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RNS</font>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#T22" target="_self">Th22</a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS</font></span>)</span> = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RNS</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RNS</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E6:123"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RNS</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>
<div><i><font color="Green">E110</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="123_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>;<br/> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">CNS</font> = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ncfcont1.html#T24" target="_self">Th24</a>, <a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E2:123_2"/>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T19">RLVECT_1:19</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="123_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>, <font color="Maroon">x2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>;<br/><b>thus </b><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font></span>)</span> = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:123_4"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RNS</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:123_4"/><i><font color="Green">E113</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">x1</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i>, <a class="ref" href="ncfcont1.html#D4" target="_self">Def4</a></i>;<br/><b>take </b><font color="Maroon">s</font> = <font color="Maroon">s</font>;<br/><b>thus </b><a NAME="E4:123_4"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>;<br/><b>assume </b><a NAME="E5:123_4"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RNS</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> )
 ;<br/><b>set </b><font color="Maroon">y</font> = <font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font>;<br/><i><font color="Green">E115</font></i>: <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x0</font> = 
<font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">RNS</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/><b>then </b><i><font color="Green">E116</font></i>: <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</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">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i/>
;<br/><a NAME="E8:123_4"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T27" target="_self">Th27</a>, <a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>;<br/><b>hence </b><a NAME="E9:123_4"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T23" target="_self">Th23</a>, , <a class="ref" href="ncfcont1.html#T26" target="_self">Th26</a>, <a class="ref" href="ncfcont1.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:123"/>
<font color="Maroon">f</font> <a href="ncfcont1.html#R9" target="_self">is_continuous_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RNS</font>
 <b>by </b><i><a class="ref" href="ncfcont1.html#T23" target="_self">Th23</a>, <a class="ref" href="ncfcont1.html#T11" target="_self">Th11</a></i>;<br/>


</div>
