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

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="borsuk_2.html#M1" target="_self">Path</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>given </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:43_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) ) ) ) )
 ;<br/>

<a NAME="E2:43_1_1"/>
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T11">TREAL_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span>;<br/>
<a NAME="E5:43_1_1"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E6:43_1_1"/><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E8:43_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="borsuk_2.html#E5:43_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E9:43_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:43_1_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:43_1_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E3:43_1_1_1_1"/><i><font color="Green">E29</font></i>: 
( 0,1 <a href="treal_1.html#K2">(#)</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="treal_1.html#K1">(#)</a> 0,1 <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a>, <a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="borsuk_2.html#E3:43_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="treal_1.html#T10">TREAL_1:10</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:43_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:43_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:43_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( <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="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span> &amp; 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E6:43_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="borsuk_2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E29</font></i>: <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,0</span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,1</span><a href="domain_1.html#K1">]</a></span>

;<br/>
<i><font color="Green">E30</font></i>: <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,1</span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E3:43_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="borsuk_1.html#K22">1[01]</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,0</span><a href="domain_1.html#K1">]</a></span>

;<br/>
<a NAME="E6:43_1_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E7:43_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,1</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E3:43_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:43_1_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,0</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E3:43_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="borsuk_2.html#E6:43_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,0 = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,0</span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,0</span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E8:43_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,1
<b>by </b><i><a class="txt" href="borsuk_2.html#E4:43_1_1_1"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green">E22</font></i></a></i>
;<br/>
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,1 = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,1</span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,1</span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E7:43_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,0
<b>by </b><i><a class="txt" href="borsuk_2.html#E5:43_1_1_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green">E22</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:43_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_2.html#E9:43_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:43_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:43_1_1_2"/><i><font color="Green">E27</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 <a href="topmetr.html#K5">I[01]</a> 
 
;<br/>
<a NAME="E2:43_1_1_2"/><i><font color="Green">E28</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E4:43_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:43_1_1_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E3:43_1_1_2_1"/><i><font color="Green">E30</font></i>: 
( 0,1 <a href="treal_1.html#K2">(#)</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="treal_1.html#K1">(#)</a> 0,1 <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a>, <a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="borsuk_2.html#E3:43_1_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="treal_1.html#T10">TREAL_1:10</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:43_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:43_1_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font>
 
;<br/>
<a NAME="E6:43_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:43_1_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span> &amp; <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="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:43_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="borsuk_2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E31</font></i>: <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E5:43_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<i><font color="Green">E32</font></i>: <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E5:43_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_2.html#E18"><i><font color="Green">Lemma14</font></i></a></i>;<br/>
<a NAME="E11:43_1_1_2"/><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E12:43_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:43_1_1_2"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="borsuk_2.html#E11:43_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green">E36</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E13:43_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E8:43_1_1_2"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green">E22</font></i></a></i>
;<br/>
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_2.html#K3" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E12:43_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E9:43_1_1_2"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green">E22</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E16:43_1_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_2.html#E14:43_1_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:43_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="borsuk_2.html#E8:43_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="borsuk_2.html#E9:43_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>


</div>
