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<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:36_1_2"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>7</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> st <br/>( <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<font color="Olive">b<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E2:36_1_2"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>7</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> st <br/>( <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<font color="Olive">b<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E2:36_1_2_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>13</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> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E3:36_1_2_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:36_1_2_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</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> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:36_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E6:36_1_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>13</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> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E7:36_1_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E8:36_1_2_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</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> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:36_1_2"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E9:36_1_2_1"/>
<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:36_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topalg_3.html#E6:36_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>;<br/><a NAME="E10:36_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:36"><i><font color="Green">E15</font></i></a>, <a class="txt" href="topalg_3.html#E3:36_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topalg_3.html#E7:36_1_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="topalg_3.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E11:36_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E4:36_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_3.html#E8:36_1_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:36_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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