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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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> 2</span>)</span>;<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>, <font color="Maroon">c<sub>8</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>9</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> 2</span>)</span>;<br/>

















<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:24"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<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="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:24"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="topreal3.html#E5:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E8:24"/><b>then </b><i><font color="Green">E21</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> 2 <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E10:24"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> 2</span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<a NAME="E11:24"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> 2</span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</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>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E13:24"/><i><font color="Green">E24</font></i>: 
 <a href="euclid.html#K14">Euclid</a> 2 <a href="hidden.html#R1">=</a>  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> 2</span>)</span> #)
 
<b>by </b><i><a class="ref" href="euclid.html#D7">EUCLID:def 7</a></i>;<br/>
<a NAME="E14:24"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>17</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>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="txt" href="topreal3.html#E1:24"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topreal3.html#E11:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E15:24"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="topreal3.html#E1:24"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topreal3.html#E8:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topreal3.html#E10:24"><i><font color="Green">E22</font></i></a>, <a class="txt" href="topreal3.html#E13:24"><i><font color="Green">E24</font></i></a>, <a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E16:24"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D5">EUCLID:def 5</a></i>;<br/>
<a NAME="E17:24"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="topreal3.html#E2:24"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topreal3.html#E3:24"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topreal3.html#E4:24"><i><font color="Green">E19</font></i></a>, <a class="ref" href="euclid.html#D16">EUCLID:def 16</a></i>;<br/>
<a NAME="E18:24"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/>
<a NAME="E19:24"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="topreal3.html#E17:24"><i><font color="Green">E28</font></i></a>, <a class="ref" href="topreal3.html#T10" target="_self">Th10</a></i>;<br/>
<i><font color="Green">E30</font></i>:  <a href="euclid.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> = 
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="finseqop.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D8">RVSUM_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="txt" href="topreal3.html#E19:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_2.html#T40">FINSEQ_2:40</a></i>
;<br/>
<i><font color="Green">E31</font></i>:  <a href="euclid.html#K11">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> = 
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="finseqop.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D8">RVSUM_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="txt" href="topreal3.html#E19:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_2.html#T40">FINSEQ_2:40</a></i>
;<br/>
<i><font color="Green">E32</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="topreal3.html#E20:24"><i><font color="Green">E30</font></i></a>, <a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="rvsum_1.html#D2">RVSUM_1:def 2</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>
<br/>.= 
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>

;<br/>
<a NAME="E23:24"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E24:24"/><b>then </b>
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#D2">RVSUM_1:def 2</a></i>;<br/>
<a NAME="E25:24"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E26:24"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E27:24"/><b>then </b>
<a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#D2">RVSUM_1:def 2</a></i>;<br/>
<a NAME="E28:24"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal3.html#E16:24"><i><font color="Green">E27</font></i></a>, <a class="txt" href="topreal3.html#E22:24"><i><font color="Green">E32</font></i></a>, <a class="txt" href="topreal3.html#E25:24"><i><font color="Green">E33</font></i></a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</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>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topreal3.html#E21:24"><i><font color="Green">E31</font></i></a>, <a class="ref" href="euclid.html#D5">EUCLID:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>
;<br/>
<a NAME="E30:24"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="topreal3.html#E15:24"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topreal3.html#E28:24"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E31:24"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="topreal3.html#E14:24"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E32:24"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>


</div>
