<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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>1</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:55"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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>1</sub></font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  &amp; <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> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font> ) ) )
 ;<br/>

<a NAME="E2:55"/><b>then </b><i><font color="Green">E50</font></i>: 
not  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T47" target="_self">Th47</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <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>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:55"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:55"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan2c.html#T48" target="_self">Th48</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E6:55"/><i><font color="Green">E52</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:55_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:55_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as   <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>1</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E7:55_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:55_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E8:55_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E11:55_1"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E3:55_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E5:55_1_1"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:55_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:55_1_1"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E13:55_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E14:55_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:55_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E3:55_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E5:55_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:55_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:55_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:55_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>
<a NAME="E16:55_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp;  <a href="metric_1.html#K4">dist</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">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan2c.html#E13:55_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>
<a NAME="E17:55_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E8:55_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E18:55_1"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E15:55_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E19:55_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E20:55_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="weierstr.html#D5">WEIERSTR:def 5</a></i>;<br/>
<a NAME="E21:55_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:55"><i><font color="Green">E51</font></i></a>, <a class="ref" href="weierstr.html#D8">WEIERSTR:def 8</a></i>;<br/>
<a NAME="E22:55_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E23:55_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E24:55_1"/><i><font color="Green">E58</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E25:55_1"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/>
<i><font color="Green">E60</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="txt" href="jordan2c.html#E25:55_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<a NAME="E27:55_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K12">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:55"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan2c.html#E23:55_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_2.html#T143">REAL_2:143</a></i>;<br/>
<a NAME="E28:55_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> 1
 
<b>by </b><i><a class="txt" href="jordan2c.html#E24:55_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/>
<a NAME="E29:55_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E30:55_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="binop_2.html#K8">"</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E31:55_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E32:55_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E33:55_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E34:55_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E32:55_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:55_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E31:55_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E36:55_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</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>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E26:55_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="toprns_1.html#T8">TOPRNS_1:8</a></i>;<br/>
<a NAME="E37:55_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</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>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T36">EUCLID:36</a></i>;<br/>
<a NAME="E38:55_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</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>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/>
<a NAME="E39:55_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</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"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/>
<a NAME="E40:55_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</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">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/>
<a NAME="E41:55_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/>
<b>hence </b><a NAME="E42:55_1"/>
<font color="Maroon">c<sub>11</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="jordan2c.html#E1:55"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan2c.html#E4:55_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:55"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E8:55"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</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="jordan2c.html#E6:55"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E9:55"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <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="jordan2c.html#E6:55"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan2c.html#E7:55"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E10:55"/><i><font color="Green">E56</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:55_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:55_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_1"><b>case </b></a><a NAME="E1:55_2_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:55_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:55_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:55"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan2c.html#E1:55_2_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E4:55_2_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as   <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>1</sub></font></span>)</span> ;<br/><i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><a NAME="E8:55_2_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><a NAME="E9:55_2_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ;<br/><a NAME="E12:55_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_2_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_2_1_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E3:55_2_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E5:55_2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:55_2_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:55_2_1_1_1_2"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E14:55_2_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E15:55_2_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:55_2_1_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_2_1_1_1_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E3:55_2_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E5:55_2_1_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:55_2_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:55_2_1_1_1_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:55_2_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/><a NAME="E17:55_2_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp;  <a href="metric_1.html#K4">dist</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">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E14:55_2_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/><a NAME="E18:55_2_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:55_2_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E19:55_2_1_1_1"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E16:55_2_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E20:55_2_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>;<br/><a NAME="E21:55_2_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="weierstr.html#D5">WEIERSTR:def 5</a></i>;<br/><a NAME="E22:55_2_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:55"><i><font color="Green">E51</font></i></a>, <a class="ref" href="weierstr.html#D8">WEIERSTR:def 8</a></i>;<br/><a NAME="E23:55_2_1_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/><a NAME="E24:55_2_1_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:55"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/><a NAME="E26:55_2_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_2_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan2c.html#E6:55_2_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan2c.html#E24:55_2_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <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>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E28:55_2_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:55"><i><font color="Green">E49</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <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>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E30:55_2_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:55"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan2c.html#E8:55"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E31:55_2_1_1_1"/><i><font color="Green">E68</font></i>: 
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E32:55_2_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1_1_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2_1_1_1_5_1"/>
( 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_1_5_1_1"><b>case </b></a><a NAME="E1:55_2_1_1_1_5_1_1"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_2_1_1_1_5_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E28:55_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:55_2_1_1_1_5_1_1"/>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_2_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan2c.html#E31:55_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma5</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span>

;<br/><a NAME="E5:55_2_1_1_1_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E26:55_2_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan2c.html#E2:55_2_1_1_1_5_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:55_2_1_1_1_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E32:55_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:55_2_1_1_1_5_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="toprns_1.html#T8">TOPRNS_1:8</a></i>;<br/><a NAME="E8:55_2_1_1_1_5_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><b>hence </b><a NAME="E9:55_2_1_1_1_5_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_1_5_1_2"><b>case </b></a><a NAME="E1:55_2_1_1_1_5_1_2"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:55_2_1_1_1_5_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:55_2_1_1_1_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/><b>hence </b><a NAME="E5:55_2_1_1_1_5_1_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E30:55_2_1_1_1"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E34:55_2_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1_2"><b>case </b></a><a NAME="E1:55_2_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_2_1_1_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/><b>hence </b><a NAME="E3:55_2_1_1_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:55_2_1_1_2"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:55_2"/>
<font color="Maroon">c<sub>11</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="jordan2c.html#E1:55"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan2c.html#E4:55_2"><i><font color="Green">E57</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:55"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:55_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:55_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_3_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1_1"><b>case </b></a><a NAME="E1:55_3_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:55_3_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:55_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:55"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan2c.html#E1:55_3_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E4:55_3_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as   <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>1</sub></font></span>)</span> ;<br/><i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><a NAME="E8:55_3_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><a NAME="E9:55_3_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ;<br/><a NAME="E12:55_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_3_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_3_1_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E3:55_3_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E5:55_3_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:55_3_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:55_3_1_1_1_2"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E14:55_3_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E15:55_3_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:55_3_1_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_3_1_1_1_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E3:55_3_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E5:55_3_1_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:55_3_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:55_3_1_1_1_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:55_3_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/><a NAME="E17:55_3_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp;  <a href="metric_1.html#K4">dist</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">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E14:55_3_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/><a NAME="E18:55_3_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:55_3_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E19:55_3_1_1_1"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E16:55_3_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E20:55_3_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>;<br/><a NAME="E21:55_3_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="weierstr.html#D5">WEIERSTR:def 5</a></i>;<br/><a NAME="E22:55_3_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:55"><i><font color="Green">E51</font></i></a>, <a class="ref" href="weierstr.html#D8">WEIERSTR:def 8</a></i>;<br/><a NAME="E23:55_3_1_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/><a NAME="E24:55_3_1_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:55"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/><a NAME="E26:55_3_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_3_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan2c.html#E6:55_3_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan2c.html#E24:55_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <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>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E28:55_3_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:55"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E29:55_3_1_1_1"/><i><font color="Green">E67</font></i>: 
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E30:55_3_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1_1_1_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_3_1_1_1_5_1"/>
( 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1_1_5_1_1"><b>case </b></a><a NAME="E1:55_3_1_1_1_5_1_1"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_3_1_1_1_5_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E28:55_3_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:55_3_1_1_1_5_1_1"/>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_3_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan2c.html#E29:55_3_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma5</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span>

;<br/><a NAME="E5:55_3_1_1_1_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E26:55_3_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan2c.html#E2:55_3_1_1_1_5_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:55_3_1_1_1_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E30:55_3_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:55_3_1_1_1_5_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="toprns_1.html#T8">TOPRNS_1:8</a></i>;<br/><a NAME="E8:55_3_1_1_1_5_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><b>hence </b><a NAME="E9:55_3_1_1_1_5_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1_1_5_1_2"><b>case </b></a><a NAME="E1:55_3_1_1_1_5_1_2"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:55_3_1_1_1_5_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:55_3_1_1_1_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><i><font color="Green">E69</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <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>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:55_3_1_1_1_5_1_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:55"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan2c.html#E9:55"><i><font color="Green">E55</font></i></a></i>;<br/><b>thus </b><a NAME="E7:55_3_1_1_1_5_1_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:55_3_1_1_1_5_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan2c.html#E6:55_3_1_1_1_5_1_2"><i><font color="Green">E70</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E32:55_3_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1_2"><b>case </b></a><a NAME="E1:55_3_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:55_3_1_1_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/><b>hence </b><a NAME="E3:55_3_1_1_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:55_3_1_1_2"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:55_3"/>
<font color="Maroon">c<sub>11</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="jordan2c.html#E1:55"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan2c.html#E4:55_3"><i><font color="Green">E57</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:55"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <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>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:55"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan2c.html#E7:55"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan2c.html#E10:55"><i><font color="Green">E56</font></i></a></i>;<br/>


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