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<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="topalg_2.html#V1" target="_self">convex</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="topalg_2.html#K1" target="_self">ConvexHomotopy</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</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/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>7</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="E1:7"/><i><font color="Green">E9</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<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>13</sub></font> =  <a href="topalg_2.html#K1" target="_self">ConvexHomotopy</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</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>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E3:7"/>
( <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>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topalg_2.html#E1:7"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E5:7"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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>16</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</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>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/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>17</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="E7:7"/><i><font color="Green">E10</font></i>: 
( <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> &amp; <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>17</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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = <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="euclid.html#K18">*</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = <font color="Maroon">a<sub>2</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<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 <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:7"/><i><font color="Green">E11</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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><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>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> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<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#S4">BINOP_1:sch 4</a>();<br/></i>
<b>consider </b><font color="Maroon">c<sub>21</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 <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E11:7"/><i><font color="Green">E12</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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><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#K2">.</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="Maroon">H<sub>2</sub></font>(<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#S4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E12:7"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,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/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</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"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E14:7"/><i><font color="Green">E13</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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><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="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 
<b>by </b><i><a class="txt" href="topalg_2.html#E9:7"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E15:7"/>
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 <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><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>23</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="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 
<b>by </b><i><a class="txt" href="topalg_2.html#E11:7"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E16:7"/><b>then </b>
( <font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <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#E14:7"><i><font color="Green">E13</font></i></a>, <a class="ref" href="topalg_1.html#T18">TOPALG_1:18</a>, <a class="ref" href="topalg_1.html#T19">TOPALG_1:19</a></i>;<br/>
<a NAME="E17:7"/><b>then </b><i><font color="Green">E14</font></i>: 
( <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> &amp; <font color="Maroon">c<sub>23</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>17</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#E7:7"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E18:7"/><i><font color="Green">E15</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>13</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>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> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</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>17</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="7_1"><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>consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</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="E2:7_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>24</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>25</sub></font>,<font color="Maroon">c<sub>26</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="E3:7_1"/><i><font color="Green">E17</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="E4:7_1"/><i><font color="Green">E18</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</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="E5:7_1"/><i><font color="Green">E19</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>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<i><font color="Green">E20</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>24</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>24</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>25</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_2.html#E2:7_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topalg_2.html#E3:7_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topalg_2.html#E5:7_1"><i><font color="Green">E19</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>26</sub></font></span>)</span> <a href="euclid.html#K18">*</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>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E9:7"><i><font color="Green">E11</font></i></a></i>
;<br/>
<i><font color="Green">E21</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</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>7</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>24</sub></font> = 
<font color="Maroon">c<sub>23</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>7</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>24</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>23</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_2.html#E2:7_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topalg_2.html#E4:7_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topalg_2.html#E5:7_1"><i><font color="Green">E19</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>26</sub></font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E11:7"><i><font color="Green">E12</font></i></a></i>
;<br/>
<b>reconsider </b><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>25</sub></font>, <font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E9:7_1"/>
<span class="p1">(<span class="default"><a href="topalg_2.html#K1" target="_self">ConvexHomotopy</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</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>26</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E10:7_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</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>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> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</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>17</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#E2:7_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topalg_2.html#E6:7_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_2.html#E7:7_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>24</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>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E20:7"/><i><font color="Green">E16</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>24</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>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> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</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>17</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>
 <b>and </b><br/><a NAME="E21:7"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>24</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_2.html#E17:7"><i><font color="Green">E14</font></i></a>, <a class="ref" href="jgraph_6.html#T20">JGRAPH_6:20</a></i>;<br/>
<a NAME="E22:7"/>
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>13</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>24</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="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</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>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> = 
<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>25</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</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>17</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>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_2.html#E18:7"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="topalg_2.html#E20:7"><i><font color="Green">E16</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E23:7"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>24</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="E24:7"/>
 <a href="topalg_2.html#K1" target="_self">ConvexHomotopy</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_2.html#E21:7"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topmetr.html#T8">TOPMETR:8</a></i>;<br/>


</div>
