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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="nfcont_1.html#V1" target="_self">compact</a>
 <b>and </b><br/><a NAME="E4:57"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nfcont_1.html#R4" target="_self">is_continuous_on</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<i><font color="Green">E47</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:57"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E6:57"/>
<font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#R4" target="_self">is_continuous_on</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:57_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E5:57"><i><font color="Green">E47</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D12" target="_self">NFCONT_1:def 12</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:57_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:57_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#R2" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E4:57"><i><font color="Green">E46</font></i></a>, <a class="ref" href="nfcont_1.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E4:57_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#R2" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T101">RELAT_1:101</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:57"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:57"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nfcont_1.html#E3:57"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nfcont_1.html#E5:57"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nfcont_1.html#T40" target="_self">Th40</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E9:57"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E5:57"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nfcont_1.html#E8:57"><i><font color="Green">E48</font></i></a></i>;<br/>

<a NAME="E10:57"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E8:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>;<br/>
<a NAME="E11:57"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E8:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="partfun2.html#T32">PARTFUN2:32</a></i>;<br/>
<a NAME="E12:57"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>
<b>hence </b><a NAME="E13:57"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E8:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nfcont_1.html#E10:57"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nfcont_1.html#E11:57"><i><font color="Green">E50</font></i></a></i>;<br/>


</div>
