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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:8"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E5:8"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<i><font color="Green">E15</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:8"><i><font color="Green">E14</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>
;<br/>
<a NAME="E7:8"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green">E17</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E5:8"><i><font color="Green">E14</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E9:8"/><i><font color="Green">E18</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E10:8"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jgraph_5.html#E3:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jgraph_5.html#E8:8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jgraph_5.html#E3:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jgraph_5.html#E8:8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<b>assume </b><a NAME="E13:8"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E15:8"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:8"><i><font color="Green">E16</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E17:8"/><i><font color="Green">E22</font></i>: 
<a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="topmetr.html#T26">TOPMETR:26</a></i>;<br/>
<a NAME="E18:8"/><i><font color="Green">E23</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jgraph_5.html#E13:8"><i><font color="Green">E20</font></i></a>, <a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E19:8"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E17:8"><i><font color="Green">E22</font></i></a>, <a class="ref" href="jordan6.html#T4">JORDAN6:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E21:8"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:8"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E13:8"><i><font color="Green">E20</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E23:8"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E15:8"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jgraph_5.html#E17:8"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jgraph_5.html#E18:8"><i><font color="Green">E23</font></i></a>, <a class="ref" href="jordan6.html#T4">JORDAN6:4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E24:8"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E13:8"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E25:8"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E26:8"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E13:8"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E27:8"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E28:8"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jgraph_5.html#E2:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jgraph_5.html#E6:8"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jgraph_5.html#E13:8"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E29:8"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E30:8"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E31:8"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E32:8"/>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E29:8"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E33:8"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E35:8"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E13:8"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jgraph_5.html#E23:8"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jgraph_5.html#E25:8"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jgraph_5.html#E27:8"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jgraph_5.html#E31:8"><i><font color="Green">E29</font></i></a>, <a class="ref" href="topreal5.html#T13">TOPREAL5:13</a></i>;<br/>
<a NAME="E36:8"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jgraph_5.html#E35:8"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E37:8"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jgraph_5.html#E35:8"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E38:8"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E36:8"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jgraph_5.html#E8:8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jgraph_5.html#E8:8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E42:8"/><i><font color="Green">E34</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E43:8"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:8"><i><font color="Green">E16</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E44:8"/>
<font color="Maroon">c<sub>19</sub></font> is   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E17:8"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jgraph_5.html#E42:8"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan6.html#T4">JORDAN6:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>17</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E46:8"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E17:8"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jgraph_5.html#E42:8"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jgraph_5.html#E43:8"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan6.html#T4">JORDAN6:4</a></i>;<br/>
<a NAME="E47:8"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E13:8"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jgraph_5.html#E22:8"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E48:8"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E25:8"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E49:8"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E9:8"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E50:8"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jgraph_5.html#E29:8"><i><font color="Green">E28</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E51:8"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E52:8"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E53:8"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jgraph_5.html#E51:8"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E54:8"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jgraph_5.html#E52:8"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E55:8"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E31:8"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jgraph_5.html#E33:8"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jgraph_5.html#E49:8"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E57:8"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E46:8"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jgraph_5.html#E50:8"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jgraph_5.html#E53:8"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jgraph_5.html#E54:8"><i><font color="Green">E42</font></i></a>, <a class="ref" href="topreal5.html#T12">TOPREAL5:12</a></i>;<br/>
<a NAME="E58:8"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jgraph_5.html#E57:8"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E59:8"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E57:8"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E60:8"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E57:8"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E61:8"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E35:8"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="txt" href="jgraph_5.html#E35:8"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E57:8"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jgraph_5.html#E60:8"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E63:8"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E6:8"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jgraph_5.html#E10:8"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jgraph_5.html#E35:8"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jgraph_5.html#E38:8"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jgraph_5.html#E57:8"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jgraph_5.html#E59:8"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div>
