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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:33"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:33"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>2</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</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="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) )
 
<b>by </b><i><a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>2</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>4</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="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) )
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><a NAME="E1:33_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E2:33"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nfcont_1.html#E3:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:33_1"/><i><font color="Green">E31</font></i>: 
<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>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1">VFUNCT_1:def 1</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E3:33_1"/><i><font color="Green">E32</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</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>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><a NAME="E4:33_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1">VFUNCT_1:def 1</a></i>;<br/><a NAME="E5:33_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1">VFUNCT_1:def 1</a></i>;<br/><a NAME="E6:33_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:33_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E3:33_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:33_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</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="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:33"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nfcont_1.html#E3:33_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E9:33_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1">VFUNCT_1:def 1</a></i>;<br/><a NAME="E10:33_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:33_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E3:33_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:33_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>4</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="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:33"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nfcont_1.html#E3:33_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E13:33_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E8:33_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="normsp_1.html#T34">NORMSP_1:34</a></i>;<br/><b>hence </b><a NAME="E14:33_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E4:33_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nfcont_1.html#T19" target="_self">Th19</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:33_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="vfunct_1.html#D1">VFUNCT_1:def 1</a></i>
<br/>.= 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="normsp_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E8:33_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nfcont_1.html#E12:33_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="normsp_1.html#T42">NORMSP_1:42</a></i>
<br/>.= 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E4:33_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nfcont_1.html#T19" target="_self">Th19</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:33"/>
<font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><a NAME="E1:33_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E2:33"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nfcont_1.html#E3:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:33_2"/><i><font color="Green">E31</font></i>: 
<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>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D2">VFUNCT_1:def 2</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E3:33_2"/><i><font color="Green">E32</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</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>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><a NAME="E4:33_2"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D2">VFUNCT_1:def 2</a></i>;<br/><a NAME="E5:33_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D2">VFUNCT_1:def 2</a></i>;<br/><a NAME="E6:33_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:33_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E3:33_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:33_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>3</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="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:33"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nfcont_1.html#E3:33_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E9:33_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D2">VFUNCT_1:def 2</a></i>;<br/><a NAME="E10:33_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:33_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E3:33_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:33_2"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>4</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="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:33"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nfcont_1.html#E3:33_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E13:33_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E8:33_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="normsp_1.html#T35">NORMSP_1:35</a></i>;<br/><b>hence </b><a NAME="E14:33_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E4:33_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nfcont_1.html#T19" target="_self">Th19</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:33_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="vfunct_1.html#D2">VFUNCT_1:def 2</a></i>
<br/>.= 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E8:33_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nfcont_1.html#E12:33_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="normsp_1.html#T43">NORMSP_1:43</a></i>
<br/>.= 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nfcont_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E4:33_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nfcont_1.html#T19" target="_self">Th19</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:33"/>
<font color="Maroon">c<sub>3</sub></font> <a href="vfunct_1.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="nfcont_1.html#R1" target="_self">is_continuous_in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D9" target="_self">Def9</a></i>;<br/>


</div>
