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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="topalg_2.html#V3" target="_self">convex</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topalg_2.html#K3" target="_self">R^1</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="topalg_2.html#K4" target="_self">R1Homotopy</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span>;<br/>
<a NAME="E1:34"/><i><font color="Green">E24</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="topalg_2.html#K4" target="_self">R1Homotopy</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3" target="_self">R^1</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E3:34"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3" target="_self">R^1</a>  <b>by </b><i><a class="txt" href="topalg_2.html#E1:34"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E5:34"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D3">BORSUK_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="ref" href="topalg_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span>;<br/>
<a NAME="E7:34"/><i><font color="Green">E25</font></i>: 
( <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T12">BORSUK_2:12</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> ,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> ,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E8:34"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> ;<br/><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>



<a NAME="E1:34_1"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:34_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <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="topalg_2.html#K3" target="_self">R^1</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>,the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> <b> such that </b><br/><a NAME="E10:34"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="topalg_2.html#E8:34"><i><font color="Green">E26</font></i></a>);<br/></i>
<a NAME="E11:34"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> ;<br/><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>



<a NAME="E1:34_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:34_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <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="topalg_2.html#K3" target="_self">R^1</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>,the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> <b> such that </b><br/><a NAME="E13:34"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="topalg_2.html#E11:34"><i><font color="Green">E28</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topalg_2.html#K3" target="_self">R^1</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3" target="_self">R^1</a>  <b>by </b><i><a class="txt" href="topalg_2.html#E5"><i><font color="Green">Lemma5</font></i></a></i>;<br/>
<a NAME="E15:34"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">c<sub>21</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E10:34"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E16:34"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3" target="_self">R^1</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_2.html#E13:34"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E17:34"/><b>then </b>
( <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i><a class="txt" href="topalg_2.html#E15:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topalg_2.html#T6" target="_self">Th6</a>, <a class="ref" href="topalg_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E18:34"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i><a class="txt" href="topalg_2.html#E7:34"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E19:34"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</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> holds <font color="Maroon">c<sub>12</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K1">*</a> <span class="p3"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K1">*</a> <span class="p3"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_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"><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>consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E2:34_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<a NAME="E3:34_3"/><i><font color="Green">E34</font></i>: 
 <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="E4:34_3"/><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</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="E5:34_3"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> = 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="topalg_2.html#E2:34_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="topalg_2.html#E3:34_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topalg_2.html#E5:34_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="topalg_2.html#E6"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E10:34"><i><font color="Green">E27</font></i></a></i>
;<br/>
<i><font color="Green">E38</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="topalg_2.html#E2:34_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="topalg_2.html#E4:34_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="topalg_2.html#E5:34_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="topalg_2.html#E6"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E13:34"><i><font color="Green">E29</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3" target="_self">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E9:34_3"/>
<span class="p1">(<span class="default"><a href="topalg_2.html#K4" target="_self">R1Homotopy</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E10:34_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K1">*</a> <span class="p3"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K1">*</a> <span class="p3"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_2.html#E2:34_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="topalg_2.html#E6:34_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topalg_2.html#E7:34_3"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>23</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>,<a href="topalg_2.html#K3" target="_self">R^1</a> <b> such that </b><br/><a NAME="E21:34"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</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><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>23</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="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">b<sub>3</sub></font>
 <b>and </b><br/><a NAME="E22:34"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_2.html#E18:34"><i><font color="Green">E31</font></i></a>, <a class="ref" href="jgraph_2.html#T29">JGRAPH_2:29</a></i>;<br/>
<a NAME="E23:34"/>
for <font color="Olive">b<sub>1</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> holds <font color="Maroon">c<sub>12</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>23</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="34_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be   <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>;<br/>

<b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K1">*</a> <span class="p3"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K1">*</a> <span class="p3"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E19:34"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="topalg_2.html#E21:34"><i><font color="Green">E33</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E24:34"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E25:34"/>
 <a href="topalg_2.html#K4" target="_self">R1Homotopy</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_2.html#E22:34"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topmetr.html#T8">TOPMETR:8</a></i>;<br/>


</div>
