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<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>, <font color="Maroon">c<sub>4</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>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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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>assume </b><b>that </b><br/><a NAME="E1:104"/><i><font color="Green">E83</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> 
 <b>and </b><br/><a NAME="E2:104"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="borsuk_6.html#R1" target="_self">are_connected</a> 
 <b>and </b><br/><a NAME="E3:104"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>and </b><br/><a NAME="E4:104"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>9</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="E6:104"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E3:104"><i><font color="Green">E85</font></i></a>, <a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</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="E8:104"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</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>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:104"><i><font color="Green">E86</font></i></a>, <a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="borsuk_6.html#K1" target="_self">L[01]</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,0,1 as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_6.html#T38" target="_self">Th38</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="borsuk_6.html#K1" target="_self">L[01]</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,0,1 as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_6.html#T38" target="_self">Th38</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>11</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>14</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>12</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>15</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/>
<a NAME="E11:104"/><i><font color="Green">E89</font></i>: 
( 0 is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 1 is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 1 <a href="real_1.html#K6">/</a> 2 is   <a href="pre_topc.html#NM2">Point</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>16</sub></font> = <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span><a href="mcart_1.html#K12">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <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="borsuk_6.html#T13" target="_self">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span><a href="mcart_1.html#K12">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <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="txt" href="borsuk_6.html#E11:104"><i><font color="Green">E89</font></i></a>, <a class="ref" href="borsuk_6.html#T13" target="_self">Th13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>18</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> <font color="Maroon">c<sub>16</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>19</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> <font color="Maroon">c<sub>17</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>11</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> 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"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<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="borsuk_2.html#T12">BORSUK_2:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_6.html#E11:104"><i><font color="Green">E89</font></i></a>, <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_6.html#E11:104"><i><font color="Green">E89</font></i></a>, <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_6.html#E11:104"><i><font color="Green">E89</font></i></a>, <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<a NAME="E18:104"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<a NAME="E19:104"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T100">TSEP_1:100</a></i>;<br/>
<a NAME="E20:104"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T31">TOPMETR:31</a></i>;<br/>
<a NAME="E21:104"/><b>then </b>
<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> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<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#E19:104"><i><font color="Green">E90</font></i></a>, <a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>20</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> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="borsuk_6.html#E6:104"><i><font color="Green">E87</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>25</sub></font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>12</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> 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"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<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="borsuk_2.html#T12">BORSUK_2:12</a></i>;<br/>
<a NAME="E24:104"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1 <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T31">TOPMETR:31</a></i>;<br/>
<a NAME="E25:104"/><b>then </b>
<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> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>,<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#E19:104"><i><font color="Green">E90</font></i></a>, <a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>25</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> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="borsuk_6.html#E8:104"><i><font color="Green">E88</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E27:104"/><i><font color="Green">E91</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> <font color="Maroon">c<sub>16</sub></font></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> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <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="borsuk_6.html#T32" target="_self">Th32</a></i>;<br/>
<i><font color="Green">E92</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> = 
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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>,<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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</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/>
<i><font color="Green">E93</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> = 
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="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></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/>
<i><font color="Green">E94</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> = 
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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>,<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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</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/>
<a NAME="E31:104"/><i><font color="Green">E95</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> <font color="Maroon">c<sub>16</sub></font></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> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>24</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>26</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="104_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="E1:104_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> <font color="Maroon">c<sub>16</sub></font></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> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:104_4"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T33" target="_self">Th33</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:104_4"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E5:104_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:104_4"><i><font color="Green">E96</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>29</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_6.html#E4:104_4"><i><font color="Green">E96</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E7:104_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:104_4"><i><font color="Green">E96</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:104_4"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E7:104_4"><i><font color="Green">E98</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:104_4"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>28</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<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="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E10:104_4"/><b>then </b>
<font color="Maroon">c<sub>27</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 <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></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="txt" href="borsuk_6.html#E4:104_4"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E5:104_4"><i><font color="Green">E97</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:104_4"/><b>then </b>
<font color="Maroon">c<sub>27</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"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<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="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E12:104_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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:104_4"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E7:104_4"><i><font color="Green">E98</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E14:104_4"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>28</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E15:104_4"/><b>then </b>
<font color="Maroon">c<sub>27</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 <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</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="txt" href="borsuk_6.html#E4:104_4"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E5:104_4"><i><font color="Green">E97</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E16:104_4"/><b>then </b>
<font color="Maroon">c<sub>27</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"><span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>,<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="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E17:104_4"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:104_4"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>28</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#E9:104_4"><i><font color="Green">E99</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E19:104_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E5:104_4"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:104_4"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <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>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:104_4"><i><font color="Green">E103</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E21:104_4"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E14:104_4"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:104_4"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <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>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E19:104_4"><i><font color="Green">E104</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E23:104_4"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E24:104_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T37" target="_self">Th37</a></i>;<br/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<font color="Maroon">c<sub>9</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="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E12:104_4"><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>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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>30</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:104_4"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E20:104_4"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E7:104_4"><i><font color="Green">E98</font></i></a>, <a class="txt" href="borsuk_6.html#E23:104_4"><i><font color="Green">E107</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:104"><i><font color="Green">E87</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:104"><i><font color="Green">E88</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</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>30</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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>30</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E7:104_4"><i><font color="Green">E98</font></i></a>, <a class="txt" href="borsuk_6.html#E24:104_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E22:104_4"><i><font color="Green">E106</font></i></a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">c<sub>26</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#E4:104_4"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E17:104_4"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E26:104_4"/>
<font color="Maroon">c<sub>24</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>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>27</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="E33:104"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>26</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#E27:104"><i><font color="Green">E91</font></i></a>, <a class="txt" href="borsuk_6.html#E31:104"><i><font color="Green">E95</font></i></a>, <a class="ref" href="borsuk_2.html#T1">BORSUK_2:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>27</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a><br/>

<a NAME="E34:104"/><i><font color="Green">E97</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>27</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"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</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>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <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="104_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:104_5"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_5_1_1"/>
( <font color="Maroon">c<sub>28</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>28</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_1_1_1"><b>suppose </b></a><a NAME="E1:104_5_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>28</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="E2:104_5_1_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E3:104_5_1_1_1"/>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:104_5_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>28</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E5:104_5_1_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</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>26</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E28:104"><i><font color="Green">E92</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:104_5_1_1_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104_5_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E7:104_5_1_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
( <font color="Maroon">c<sub>28</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> &amp; 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>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E8:104_5_1_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</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>20</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E29:104"><i><font color="Green">E93</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:104_5_1_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>28</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#E7:104_5_1_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E10:104_5_1_1_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E7:104_5_1_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:104_5_1_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <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>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E9:104_5_1_1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><i><font color="Green">E106</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104_5_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="borsuk_6.html#T39" target="_self">Th39</a></i>
<br/>.= 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font>

;<br/><a NAME="E13:104_5_1_1_1"/><i><font color="Green">E107</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font> 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#E1:104_5_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_6.html#T6" target="_self">Th6</a></i>;<br/><font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,0 = 
<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>28</sub></font>,0</span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>24</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>28</sub></font>,0</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:104"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E5:104_5_1_1_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</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="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,0</span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:104_5_1_1_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E11:104_5_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,0</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E11:104"><i><font color="Green">E89</font></i></a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,0
<b>by </b><i><a class="txt" href="borsuk_6.html#E12:104_5_1_1_1"><i><font color="Green">E106</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <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>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:104"><i><font color="Green">E87</font></i></a>, <a class="txt" href="borsuk_6.html#E13:104_5_1_1_1"><i><font color="Green">E107</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:104_5_1_1_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104"><i><font color="Green">E83</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104"><i><font color="Green">E84</font></i></a>, <a class="txt" href="borsuk_6.html#E1:104_5_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_1_1_2"><b>suppose </b></a><a NAME="E1:104_5_1_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:104_5_1_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E3:104_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:104_5_1_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>28</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span> &amp; 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>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E5:104_5_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</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>26</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E28:104"><i><font color="Green">E92</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:104_5_1_1_2"/><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</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>25</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E30:104"><i><font color="Green">E94</font></i></a>, <a class="txt" href="borsuk_6.html#E4:104_5_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:104_5_1_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:104_5_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:104_5_1_1_2"/>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:104_5_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:104_5_1_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <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>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E7:104_5_1_1_2"><i><font color="Green">E104</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><i><font color="Green">E106</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104_5_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="borsuk_6.html#T39" target="_self">Th39</a></i>
<br/>.= 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/><a NAME="E11:104_5_1_1_2"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 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#E1:104_5_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_6.html#T7" target="_self">Th7</a></i>;<br/><font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,0 = 
<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>28</sub></font>,0</span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>26</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>28</sub></font>,0</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:104"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E5:104_5_1_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,0</span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:104_5_1_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E9:104_5_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,0</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E11:104"><i><font color="Green">E89</font></i></a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="binop_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>28</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,0
<b>by </b><i><a class="txt" href="borsuk_6.html#E10:104_5_1_1_2"><i><font color="Green">E106</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</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>28</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#E8:104"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E11:104_5_1_1_2"><i><font color="Green">E107</font></i></a></i>
;<br/><b>hence </b><a NAME="E13:104_5_1_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104"><i><font color="Green">E83</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104"><i><font color="Green">E84</font></i></a>, <a class="txt" href="borsuk_6.html#E1:104_5_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:104_5"/>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_5_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_5_2_1"/>
( <font color="Maroon">c<sub>28</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>28</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_2_1_1"><b>suppose </b></a><a NAME="E1:104_5_2_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>28</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="E2:104_5_2_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E3:104_5_2_1_1"/>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:104_5_2_1_1"/><b>then </b>
not <font color="Maroon">c<sub>28</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E5:104_5_2_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</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>26</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E28:104"><i><font color="Green">E92</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:104_5_2_1_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104_5_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E7:104_5_2_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
( <font color="Maroon">c<sub>28</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> &amp; 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>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E8:104_5_2_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</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>20</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E29:104"><i><font color="Green">E93</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:104_5_2_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>28</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#E7:104_5_2_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E10:104_5_2_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E7:104_5_2_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:104_5_2_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <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>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E9:104_5_2_1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><i><font color="Green">E106</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104_5_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="borsuk_6.html#T39" target="_self">Th39</a></i>
<br/>.= 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font>

;<br/><a NAME="E13:104_5_2_1_1"/><i><font color="Green">E107</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font> 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#E1:104_5_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_6.html#T6" target="_self">Th6</a></i>;<br/><font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,1 = 
<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>28</sub></font>,1</span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>24</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>28</sub></font>,1</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:104"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E5:104_5_2_1_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</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="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,1</span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:104_5_2_1_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E11:104_5_2_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E11:104"><i><font color="Green">E89</font></i></a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,1
<b>by </b><i><a class="txt" href="borsuk_6.html#E12:104_5_2_1_1"><i><font color="Green">E106</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <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>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:104"><i><font color="Green">E87</font></i></a>, <a class="txt" href="borsuk_6.html#E13:104_5_2_1_1"><i><font color="Green">E107</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:104_5_2_1_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104"><i><font color="Green">E83</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104"><i><font color="Green">E84</font></i></a>, <a class="txt" href="borsuk_6.html#E1:104_5_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_5_2_1_2"><b>suppose </b></a><a NAME="E1:104_5_2_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:104_5_2_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E3:104_5_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_2_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:104_5_2_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>28</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"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span> &amp; 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>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E5:104_5_2_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</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>26</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E28:104"><i><font color="Green">E92</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:104_5_2_1_2"/><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</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>25</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E30:104"><i><font color="Green">E94</font></i></a>, <a class="txt" href="borsuk_6.html#E4:104_5_2_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:104_5_2_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:104_5_2_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:104_5_2_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:104_5_2_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:104_5_2_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <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>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E7:104_5_2_1_2"><i><font color="Green">E104</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><i><font color="Green">E106</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5_2_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104_5_2_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="borsuk_6.html#T39" target="_self">Th39</a></i>
<br/>.= 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/><a NAME="E11:104_5_2_1_2"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 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#E1:104_5_2_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_6.html#T7" target="_self">Th7</a></i>;<br/><font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,1 = 
<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>28</sub></font>,1</span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>26</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>28</sub></font>,1</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:104"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E5:104_5_2_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,1</span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:104_5_2_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E9:104_5_2_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E11:104"><i><font color="Green">E89</font></i></a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="binop_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>28</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>,1
<b>by </b><i><a class="txt" href="borsuk_6.html#E10:104_5_2_1_2"><i><font color="Green">E106</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>28</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#E8:104"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_6.html#E11:104_5_2_1_2"><i><font color="Green">E107</font></i></a></i>
;<br/><b>hence </b><a NAME="E13:104_5_2_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104"><i><font color="Green">E83</font></i></a>, <a class="txt" href="borsuk_6.html#E2:104"><i><font color="Green">E84</font></i></a>, <a class="txt" href="borsuk_6.html#E1:104_5_2_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:104_5"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_5"><i><font color="Green">E98</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:104"/>
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>27</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>27</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>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<i><font color="Green">E98</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> = 
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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>,<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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</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/>
<a NAME="E2:104_6"/>
not 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:104_6"/><b>then </b>
not 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E4:104_6"/><b>then </b><i><font color="Green">E99</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>28</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>26</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_6"><i><font color="Green">E98</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:104_6"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E6:104_6"/><b>then </b><i><font color="Green">E100</font></i>: 
1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E7:104_6"/><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>28</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>26</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:104_6"><i><font color="Green">E98</font></i></a>, <a class="txt" href="borsuk_6.html#E6:104_6"><i><font color="Green">E100</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:104_6"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:104_6"/><b>then </b><i><font color="Green">E102</font></i>: 
0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<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="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E10:104_6"/><b>then </b><i><font color="Green">E103</font></i>: 
0 <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="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:104_6"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
;<br/>
<a NAME="E12:104_6"/><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>28</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>20</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E29:104"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_6.html#E9:104_6"><i><font color="Green">E102</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:104_6"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E14:104_6"/><b>then </b>
1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E15:104_6"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>28</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>25</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E30:104"><i><font color="Green">E94</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E16:104_6"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E11:104_6"><i><font color="Green">E104</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:104_6"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <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>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E10:104_6"><i><font color="Green">E103</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E18:104_6"/><i><font color="Green">E108</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:104_6"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E19:104_6"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
;<br/>
<a NAME="E20:104_6"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:104_6"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <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>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E18:104_6"><i><font color="Green">E108</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>28</sub></font> = 
<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>28</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>24</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>28</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:104"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E4:104_6"><i><font color="Green">E99</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</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="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E12:104_6"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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>28</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E17:104_6"><i><font color="Green">E107</font></i></a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</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>28</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T37" target="_self">Th37</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>28</sub></font>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:104"><i><font color="Green">E87</font></i></a></i>
;<br/>

<font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>28</sub></font> = 
<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">1,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>26</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>28</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E33:104"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E7:104_6"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E15:104_6"><i><font color="Green">E106</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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>28</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E21:104_6"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</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>28</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T37" target="_self">Th37</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>28</sub></font>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E8:104"><i><font color="Green">E88</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E24:104_6"/>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E36:104"/>
( <font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( 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="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>27</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"><font color="Maroon">c<sub>5</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>7</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>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="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 <br/>( <font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E33:104"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_6.html#E34:104"><i><font color="Green">E97</font></i></a></i>;<br/>


</div>
