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<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<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>4</sub></font>;<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>4</sub></font>;<br/>





<b>assume </b><a NAME="E1:143"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="clvect_1.html#V6" target="_self">convergent</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:143"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="clvect_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="clvect_1.html#D15" target="_self">CLVECT_1:def 15</a><br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E4:143"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:143"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E3:143"><i><font color="Green">E91</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E7:143"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E8:143"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E6:143"><i><font color="Green">E92</font></i></a></i>;<br/>
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)</span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>

<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>

;<br/>
<b>hence </b><a NAME="E10:143"/>
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">c<sub>6</sub></font> <a href="clvect_1.html#K8" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E8:143"><i><font color="Green">E93</font></i></a>, <a class="ref" href="clvect_1.html#D13" target="_self">Def13</a></i>;<br/>


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