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<a NAME="E1:46_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="topalg_5.html#K8" target="_self">Ciso</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="gr_cy_1.html#D4">GR_CY_1:def 4</a></i>;<br/>
<b>thus </b><a NAME="E2:46_1_1"/>
 <a href="topalg_5.html#K8" target="_self">Ciso</a>  is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="topalg_5.html#K8" target="_self">Ciso</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="topalg_5.html#K8" target="_self">Ciso</a>  )
 <b>and </b><br/><a NAME="E2:46_1_1_1"/><i><font color="Green">E44</font></i>: 
<a href="topalg_5.html#K8" target="_self">Ciso</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <a href="topalg_5.html#K8" target="_self">Ciso</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="topalg_5.html#E1:46_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="int_1.html#D2">INT_1:def 2</a>, <a class="ref" href="gr_cy_1.html#D4">GR_CY_1:def 4</a></i>;<br/>
<a NAME="E4:46_1_1_1"/>
( <a href="topalg_5.html#K8" target="_self">Ciso</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <a href="topalg_5.html#K8" target="_self">Ciso</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="topalg_5.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:46_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>10</sub></font>, <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>11</sub></font> <a href="borsuk_2.html#R4">are_homotopic</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:46_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:46_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E8:46_1_1_1"/><i><font color="Green">E47</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>12</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"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>10</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>12</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"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>11</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="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K9">c[10]</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K9">c[10]</a>  ) ) )
 <b>by </b><i><a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a></i>;<br/>
<a NAME="E9:46_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is    <a href="borsuk_6.html#M1">Homotopy</a> of  <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>10</sub></font>, <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7:46_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topalg_5.html#E8:46_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topalg_5.html#E5:46_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="toprealb.html#K4">R^1</a> 0;<br/>
<a NAME="E10:46_1_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="toprealb.html#K4">R^1</a> 0 <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="toprealb.html#K9">c[10]</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a>, <a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="toprealb.html#T34">TOPREALB:34</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a> , <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="borsuk_2.html#M1">Path</a> of  <a href="toprealb.html#K4">R^1</a> 0,<font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E12:46_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="borsuk_2.html#R4">are_homotopic</a> 
 <b>and </b><br/><a NAME="E13:46_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E14:46_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="toprealb.html#K9">c[10]</a> </span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E15:46_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> st <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:46_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topalg_5.html#E9:46_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="topalg_5.html#T24" target="_self">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E17:46_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K4">R^1</a> 0
 <b>and </b><br/><a NAME="E18:46_1_1_1"/>
 <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E19:46_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E20:46_1_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K4">R^1</a> 0 holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:46_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topalg_5.html#T23" target="_self">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E22:46_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K4">R^1</a> 0
 <b>and </b><br/><a NAME="E23:46_1_1_1"/>
 <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E24:46_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E25:46_1_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K4">R^1</a> 0 holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:46_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topalg_5.html#T23" target="_self">Th23</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> =  <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>17</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> =  <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font>;<br/>
<i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="topalg_5.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="toprealb.html#K4">R^1</a> 0
<b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a></i>
;<br/>
<i><font color="Green">E55</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c<sub>17</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a></i>
<br/>.= 
 <a href="toprealb.html#K4">R^1</a> 0
<b>by </b><i><a class="txt" href="topalg_5.html#E12:46_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
;<br/>
<a NAME="E28:46_1_1_1"/>
 <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E29:46_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:46_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E26:46_1_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:46_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="binop_1.html#T30">BINOP_1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E8:46_1_1_1"><i><font color="Green">E47</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E31:46_1_1_1"/><b>then </b>
 <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E32:46_1_1_1"/><b>then </b>
 <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:46_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E27:46_1_1_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E33:46_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> 1
 
<b>by </b><i><a class="txt" href="topalg_5.html#E29:46_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topalg_5.html#D1" target="_self">Def1</a></i>;<br/>
<i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="topalg_5.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="toprealb.html#K4">R^1</a> 0
<b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a></i>
;<br/>
<a NAME="E35:46_1_1_1"/>
 <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E36:46_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E25:46_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topalg_5.html#E34:46_1_1_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">c<sub>17</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a></i>
<br/>.= 
 <a href="toprealb.html#K4">R^1</a> 0
<b>by </b><i><a class="txt" href="topalg_5.html#E12:46_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:46_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="binop_1.html#T30">BINOP_1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E8:46_1_1_1"><i><font color="Green">E47</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E39:46_1_1_1"/><b>then </b>
 <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E40:46_1_1_1"/><b>then </b>
 <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E25:46_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topalg_5.html#E37:46_1_1_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E41:46_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> 1
 
<b>by </b><i><a class="txt" href="topalg_5.html#E36:46_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="topalg_5.html#D1" target="_self">Def1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E12:46_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E12:46_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E43:46_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E33:46_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topalg_5.html#E41:46_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E3:46_1_1"/>
 <a href="relset_1.html#K5">rng</a> <a href="topalg_5.html#K8" target="_self">Ciso</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:46_1_1"/>
<font color="Maroon">c<sub>8</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="topalg_1.html#NK4" target="_self">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of  <a href="toprealb.html#K9">c[10]</a> <b> such that </b><br/><a NAME="E6:46_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48">TOPALG_1:48</a></i>;<br/>
<a NAME="E7:46_1_1"/>
 <a href="toprealb.html#K4">R^1</a> 0 <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="toprealb.html#K9">c[10]</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a>, <a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="toprealb.html#T34">TOPREALB:34</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E9:46_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K4">R^1</a> 0
 <b>and </b><br/><a NAME="E10:46_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E11:46_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E12:46_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K4">R^1</a> 0 holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="topalg_5.html#T23" target="_self">Th23</a></i>;<br/>
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="toprealb.html#K9">c[10]</a> 
<b>by </b><i><a class="txt" href="topalg_5.html#E10:46_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>
;<br/>
<a NAME="E14:46_1_1"/><b>then </b>
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="toprealb.html#K9">c[10]</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:46_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="toprealb.html#T34">TOPREALB:34</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E16:46_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> 1 is   <a href="int_1.html#NM1">Integer</a>
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E17:46_1_1"/>
<font color="Maroon">c<sub>10</sub></font> is    <a href="borsuk_2.html#M1">Path</a> of  <a href="toprealb.html#K4">R^1</a> 0, <a href="toprealb.html#K4">R^1</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:46_1_1"><i><font color="Green">E46</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a><br/>

<b>thus </b><a NAME="E2:46_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K4">R^1</a> 0
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:46_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>

<b>thus </b><a NAME="E3:46_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K4">R^1</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:46_1_1"/><b>then </b>
<a href="topalg_5.html#K8" target="_self">Ciso</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E16:46_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="topalg_5.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E19:46_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <a href="topalg_5.html#K8" target="_self">Ciso</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:46_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topalg_5.html#E1:46_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_5.html#E6:46_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topalg_5.html#E10:46_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div>
