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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green">E14</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:13"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="topalg_1.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topalg_1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topalg_1.html#E3:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E7:13"/><i><font color="Green">E16</font></i>: 
 <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<i><font color="Green">E17</font></i>:  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topalg_1.html#E2:13"><i><font color="Green">E12</font></i></a>, <a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topalg_1.html#E3:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E7:13"><i><font color="Green">E16</font></i></a>, <a class="ref" href="topalg_1.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14">EUCLID:14</a></i>
;<br/>
<a NAME="E9:13"/><i><font color="Green">E18</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>;<br/>
<a NAME="E10:13"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E5:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/>
<b>hence </b><a NAME="E11:13"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E4:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topalg_1.html#E8:13"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topalg_1.html#E9:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>


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