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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="funct_1.html#V3">constant</a>  <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:115"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 ;<br/>

<a NAME="E2:115"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <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>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</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 <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>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T39">YELLOW12:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</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 <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>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T40">YELLOW12:40</a></i>;<br/>
<a NAME="E5:115"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font> as  <a href="pre_topc.html#V5">continuous</a>  <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a>  as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E8:115"/>
<font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E10:115"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span> holds <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_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 <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span>;<br/>

<a NAME="E1:115_1"/>
<font color="Maroon">c<sub>16</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"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span>
 
;<br/>
<a NAME="E2:115_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK9" target="_self">IAA</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E3:115_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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>
 
;<br/>
<a NAME="E4:115_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:115_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E6:115_1"/><b>then </b><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K1">`1</a>  <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>  &amp; <font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K2">`2</a>  <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>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:115_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:115_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:115_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/>
<a NAME="E8:115_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:115_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:115_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:115_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="borsuk_6.html#T67" target="_self">Th67</a></i>;<br/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:115_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:115_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E6:115_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E7:115_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="borsuk_6.html#E9:115_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E11:115_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <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>by </b><i><a class="txt" href="borsuk_6.html#E2:115"><i><font color="Green">E89</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a>  as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</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#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E14:115"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span> holds <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>;<br/>

<a NAME="E1:115_2"/>
<font color="Maroon">c<sub>20</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"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>
 
;<br/>
<a NAME="E2:115_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E3:115_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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>
 
;<br/>
<a NAME="E4:115_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:115_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E6:115_2"/><b>then </b><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K1">`1</a>  <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>  &amp; <font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a>  <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>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:115_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:115_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:115_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>;<br/>
<a NAME="E8:115_2"/><i><font color="Green">E99</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <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> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:115_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="jordan5b.html#T4">JORDAN5B:4</a></i>;<br/>
<a NAME="E9:115_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:115_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:115_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E7:115_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="borsuk_6.html#E9:115_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E8:115_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_2.html#D6">BORSUK_2:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E11:115_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a>  as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pre_topc.html#T30">PRE_TOPC:30</a></i>;<br/>
<a NAME="E16:115"/>
<font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E18:115"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span> holds <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>;<br/>

<a NAME="E1:115_3"/>
<font color="Maroon">c<sub>23</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"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>
 
;<br/>
<a NAME="E2:115_3"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E3:115_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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>
 
;<br/>
<a NAME="E4:115_3"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:115_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E6:115_3"/><b>then </b><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K1">`1</a>  <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>  &amp; <font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K2">`2</a>  <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>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:115_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:115_3"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:115_3"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>;<br/>
<a NAME="E8:115_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:115_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:115_3"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:115_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="borsuk_6.html#T68" target="_self">Th68</a></i>;<br/>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:115_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:115_3"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E6:115_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="borsuk_6.html#E7:115_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="borsuk_6.html#E9:115_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E11:115_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>23</sub></font> = <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> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="subset_1.html#K4">\/</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>;<br/>
<a NAME="E19:115"/>
<a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="subset_1.html#K4">\/</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E20:115"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="subset_1.html#K4">\/</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <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> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="subset_1.html#K4">\/</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</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> <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E22:115"/><i><font color="Green">E94</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="subset_1.html#K4">\/</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E23:115"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="subset_1.html#K4">\/</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<a NAME="E24:115"/><i><font color="Green">E96</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="borsuk_6.html#NK9" target="_self">IAA</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E25:115"/><i><font color="Green">E97</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span>, <font color="Maroon">c<sub>26</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>24</sub></font> <b>by </b><i><a class="txt" href="borsuk_6.html#E22:115"><i><font color="Green">E94</font></i></a>, <a class="txt" href="borsuk_6.html#E24:115"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E27:115"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E23:115"><i><font color="Green">E95</font></i></a>, <a class="txt" href="borsuk_6.html#E25:115"><i><font color="Green">E97</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E28:115"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</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="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:115_4"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:115_4"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E3:115_4"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="subset_1.html#K9">/\</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="pre_topc.html#NM2">Point</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><b> such that </b><br/><a NAME="E5:115_4"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="borsuk_6.html#T65" target="_self">Th65</a></i>;<br/>
<a NAME="E6:115_4"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK9" target="_self">IAA</a>  &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:115_4"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E8:115_4"/><i><font color="Green">E103</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:115_4"><i><font color="Green">E101</font></i></a></i>;<br/>
<a NAME="E9:115_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>29</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:115_4"><i><font color="Green">E102</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E10:115"><i><font color="Green">E91</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E14:115"><i><font color="Green">E92</font></i></a>, <a class="txt" href="borsuk_6.html#E5:115_4"><i><font color="Green">E101</font></i></a>, <a class="txt" href="borsuk_6.html#E8:115_4"><i><font color="Green">E103</font></i></a>, <a class="txt" href="borsuk_6.html#E9:115_4"><i><font color="Green">E104</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:115_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E29:115"/>
 <a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="tarski.html#R1">c=</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="subset_1.html#K4">\/</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E30:115"/><b>then </b><i><font color="Green">E100</font></i>: 
<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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a>  is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T29">TOPMETR:29</a></i>;<br/>
<a NAME="E31:115"/>
 <a href="borsuk_6.html#NK11" target="_self">IBB</a>  <a href="tarski.html#R1">c=</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a>  <a href="subset_1.html#K4">\/</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E32:115"/><b>then </b><i><font color="Green">E101</font></i>: 
<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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a>  is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T29">TOPMETR:29</a></i>;<br/>
<a NAME="E33:115"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>25</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E24:115"><i><font color="Green">E96</font></i></a>, <a class="ref" href="tops_2.html#T34">TOPS_2:34</a></i>;<br/>
<a NAME="E34:115"/>
<font color="Maroon">c<sub>26</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E25:115"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tops_2.html#T34">TOPS_2:34</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E36:115"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E27:115"><i><font color="Green">E98</font></i></a>, <a class="txt" href="borsuk_6.html#E28:115"><i><font color="Green">E99</font></i></a>, <a class="txt" href="borsuk_6.html#E30:115"><i><font color="Green">E100</font></i></a>, <a class="txt" href="borsuk_6.html#E32:115"><i><font color="Green">E101</font></i></a>, <a class="txt" href="borsuk_6.html#E33:115"><i><font color="Green">E102</font></i></a>, <a class="ref" href="jgraph_2.html#T9">JGRAPH_2:9</a></i>;<br/>
<a NAME="E37:115"/><i><font color="Green">E104</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>29</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span> as   <a href="struct_0.html#NM2">Subset</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> <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<i><font color="Green">E105</font></i>: <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</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>
<b>by </b><i><a class="txt" href="borsuk_6.html#E23:115"><i><font color="Green">E95</font></i></a>, <a class="txt" href="borsuk_6.html#E37:115"><i><font color="Green">E104</font></i></a>, <a class="ref" href="borsuk_6.html#T64" target="_self">Th64</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <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>
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>
;<br/>
<a NAME="E40:115"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</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="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:115_6"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:115_6"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#NK11" target="_self">IBB</a>  <a href="subset_1.html#K9">/\</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E23:115"><i><font color="Green">E95</font></i></a>, <a class="txt" href="borsuk_6.html#E37:115"><i><font color="Green">E104</font></i></a>, <a class="ref" href="borsuk_6.html#T80" target="_self">Th80</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E3:115_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a>  <a href="subset_1.html#K9">/\</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T74" target="_self">Th74</a>, <a class="ref" href="borsuk_6.html#T75" target="_self">Th75</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:115_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a>  <a href="subset_1.html#K9">/\</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:115_6"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a>  <a href="subset_1.html#K9">/\</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:115_6"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="pre_topc.html#NM2">Point</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><b> such that </b><br/><a NAME="E7:115_6"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 <b>by </b><i><a class="ref" href="borsuk_6.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E8:115_6"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a>  &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a>  )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:115_6"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>30</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E10:115_6"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>32</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:115_6"><i><font color="Green">E110</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E11:115_6"/>
( <font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K1">`1</a>  is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; <font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K2">`2</a>  is   <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_6.html#E7:115_6"><i><font color="Green">E109</font></i></a>, <a class="ref" href="borsuk_6.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E12:115_6"/><b>then </b><i><font color="Green">E112</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <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> 
 
<b>by </b><i><a class="ref" href="jordan5b.html#T4">JORDAN5B:4</a></i>;<br/>
<a NAME="E13:115_6"/>
<font color="Maroon">c<sub>30</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"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:115_6"><i><font color="Green">E110</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E14:115_6"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> = 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E36:115"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E14:115"><i><font color="Green">E92</font></i></a>, <a class="txt" href="borsuk_6.html#E10:115_6"><i><font color="Green">E111</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E12:115_6"><i><font color="Green">E112</font></i></a>, <a class="ref" href="borsuk_2.html#D6">BORSUK_2:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:115"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_6.html#E7:115_6"><i><font color="Green">E109</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E16:115_6"/>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E41:115"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>28</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E23:115"><i><font color="Green">E95</font></i></a>, <a class="ref" href="tops_1.html#T36">TOPS_1:36</a></i>;<br/>
<a NAME="E42:115"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>30</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><br/><a NAME="E44:115"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E36:115"><i><font color="Green">E103</font></i></a>, <a class="txt" href="borsuk_6.html#E39:115"><i><font color="Green">E105</font></i></a>, <a class="txt" href="borsuk_6.html#E40:115"><i><font color="Green">E106</font></i></a>, <a class="txt" href="borsuk_6.html#E41:115"><i><font color="Green">E107</font></i></a>, <a class="ref" href="jgraph_2.html#T9">JGRAPH_2:9</a></i>;<br/>
<a NAME="E45:115"/><i><font color="Green">E109</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>30</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>30</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>5</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="115_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>thus </b><a NAME="E1:115_7"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_7_1"><b>proof </b></a><div class="add">

<a NAME="E1:115_7_1"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_7_1_1"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_7_1_1_1"><b>suppose </b></a><a NAME="E1:115_7_1_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:115_7_1_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK9" target="_self">IAA</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T78" target="_self">Th78</a></i>;<br/><a NAME="E3:115_7_1_1_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:115_7_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E4:115_7_1_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E5:115_7_1_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><a NAME="E6:115_7_1_1_1"/><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E2:115_7_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E7:115_7_1_1_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:115_7_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E8:115_7_1_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E9:115_7_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E44:115"><i><font color="Green">E108</font></i></a>, <a class="txt" href="borsuk_6.html#E5:115_7_1_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E36:115"><i><font color="Green">E103</font></i></a>, <a class="txt" href="borsuk_6.html#E9:115_7_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E10:115"><i><font color="Green">E91</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="borsuk_6.html#E6:115_7_1_1_1"><i><font color="Green">E114</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E11:115_7_1_1_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_7_1_1_2"><b>suppose </b></a><a NAME="E1:115_7_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:115_7_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T74" target="_self">Th74</a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E3:115_7_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E44:115"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:115"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="borsuk_6.html#E2:115_7_1_1_2"><i><font color="Green">E112</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E5:115_7_1_1_2"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_7_1_1_3"><b>suppose </b></a><a NAME="E1:115_7_1_1_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:115_7_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T77" target="_self">Th77</a></i>;<br/><a NAME="E3:115_7_1_1_3"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E4:115_7_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,0</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E44:115"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:115"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="borsuk_6.html#E3:115_7_1_1_3"><i><font color="Green">E112</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_1_1_3"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E6:115_7_1_1_3"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E2:115_7"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_7_2"><b>proof </b></a><div class="add">

<a NAME="E1:115_7_2"/><i><font color="Green">E110</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E2:115_7_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <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="E3:115_7_2"/><b>then </b><i><font color="Green">E111</font></i>: 
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_7_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_7_2_1"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 or <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_7_2_1_1"><b>suppose </b></a><a NAME="E1:115_7_2_1_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:115_7_2_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T71" target="_self">Th71</a></i>;<br/><a NAME="E3:115_7_2_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E4:115_7_2_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><a NAME="E5:115_7_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T73" target="_self">Th73</a></i>;<br/><a NAME="E6:115_7_2_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E7:115_7_2_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E44:115"><i><font color="Green">E108</font></i></a>, <a class="txt" href="borsuk_6.html#E4:115_7_2_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E36:115"><i><font color="Green">E103</font></i></a>, <a class="txt" href="borsuk_6.html#E7:115_7_2_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E14:115"><i><font color="Green">E92</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="borsuk_6.html#E6:115_7_2_1_1"><i><font color="Green">E113</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:115"><i><font color="Green">E90</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:115_7_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>
;<br/><b>hence </b><a NAME="E9:115_7_2_1_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_7_2_1_2"><b>suppose </b></a><a NAME="E1:115_7_2_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:115_7_2_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T74" target="_self">Th74</a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E3:115_7_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,1</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E44:115"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:115"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="borsuk_6.html#E2:115_7_2_1_2"><i><font color="Green">E113</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_7_2_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:115"><i><font color="Green">E90</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:115_7_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>
;<br/><b>hence </b><a NAME="E5:115_7_2_1_2"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E46:115"/>
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>30</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>3</sub></font> &amp; <font color="Maroon">c<sub>30</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="115_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>thus </b><a NAME="E1:115_8"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>31</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="115_8_1"><b>proof </b></a><div class="add">

<a NAME="E1:115_8_1"/>
0 <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> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>32</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span> as   <a href="pre_topc.html#NM2">Point</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> <b>by </b><i><a class="txt" href="borsuk_6.html#E95"><i><font color="Green">Lemma80</font></i></a></i>;<br/>
<a NAME="E3:115_8_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK9" target="_self">IAA</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T69" target="_self">Th69</a></i>;<br/>
<a NAME="E4:115_8_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>32</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK9" target="_self">IAA</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E5:115_8_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E6:115_8_1"/><b>then </b>
not <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T68" target="_self">Th68</a></i>;<br/>
<a NAME="E7:115_8_1"/><b>then </b>
not <font color="Maroon">c<sub>32</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"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E8:115_8_1"/><b>then </b><i><font color="Green">E112</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
 
;<br/>
<i><font color="Green">E113</font></i>: <font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>31</sub></font> = 
<font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E44:115"><i><font color="Green">E108</font></i></a>, <a class="txt" href="borsuk_6.html#E8:115_8_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_8_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_8_1_2"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 or <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_8_1_2_1"><b>suppose </b></a><a NAME="E1:115_8_1_2_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:115_8_1_2_1"/><b>then </b>
not <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK11" target="_self">IBB</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T70" target="_self">Th70</a></i>;<br/><a NAME="E3:115_8_1_2_1"/><b>then </b>
not <font color="Maroon">c<sub>32</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"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E4:115_8_1_2_1"/><b>then </b>
not <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><b>then </b><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E36:115"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E10:115"><i><font color="Green">E91</font></i></a>, <a class="txt" href="borsuk_6.html#E4:115_8_1"><i><font color="Green">E110</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E5:115_8_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
;<br/><b>hence </b><a NAME="E6:115_8_1_2_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>31</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_6.html#E9:115_8_1"><i><font color="Green">E113</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_8_1_2_2"><b>suppose </b></a><a NAME="E1:115_8_1_2_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:115_8_1_2_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>32</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"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK11" target="_self">IBB</a> </span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T72" target="_self">Th72</a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E3:115_8_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E36:115"><i><font color="Green">E103</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_8_1_2_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E14:115"><i><font color="Green">E92</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_8_1_2_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="borsuk_6.html#E2:115_8_1_2_2"><i><font color="Green">E115</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E5:115_8_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="borsuk_6.html#E1:115_8_1_2_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
;<br/><b>hence </b><a NAME="E5:115_8_1_2_2"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>31</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_6.html#E9:115_8_1"><i><font color="Green">E113</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E2:115_8"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>31</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="115_8_2"><b>proof </b></a><div class="add">

<a NAME="E1:115_8_2"/>
1 <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> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>32</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span> as   <a href="pre_topc.html#NM2">Point</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> <b>by </b><i><a class="txt" href="borsuk_6.html#E95"><i><font color="Green">Lemma80</font></i></a></i>;<br/>
<a NAME="E3:115_8_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_6.html#NK13" target="_self">ICC</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T76" target="_self">Th76</a></i>;<br/>
<a NAME="E4:115_8_2"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E5:115_8_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>32</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"><span class="p2"><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> <a href="pre_topc.html#K3">|</a> <a href="borsuk_6.html#NK13" target="_self">ICC</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E3:115_8_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E6:115_8_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E114</font></i>: <font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>31</sub></font> = 
<font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E44:115"><i><font color="Green">E108</font></i></a>, <a class="txt" href="borsuk_6.html#E6:115_8_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:115"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_6.html#E5:115_8_2"><i><font color="Green">E112</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:115"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E4:115_8_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E9:115_8_2"/>
<font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>31</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_6.html#E7:115_8_2"><i><font color="Green">E114</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E47:115"/>
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E44:115"><i><font color="Green">E108</font></i></a>, <a class="txt" href="borsuk_6.html#E45:115"><i><font color="Green">E109</font></i></a>, <a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a></i>;<br/>


</div>
