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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> 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>4</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>assume </b><a NAME="E1:63"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="borsuk_6.html#R1">are_connected</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </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="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>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E2:63_1"/><i><font color="Green">E41</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" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self">Th48</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:63_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</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" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self">Th48</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:63_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <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" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self">Th48</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:63_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</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" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48" target="_self">Th48</a></i>;<br/><a NAME="E9:63_1"/>
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <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" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:63"><i><font color="Green">E40</font></i></a>, <a class="txt" href="topalg_1.html#E2:63_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="topalg_1.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E10:63_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E6:63_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E11:63_1"/>
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</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" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:63"><i><font color="Green">E40</font></i></a>, <a class="txt" href="topalg_1.html#E4:63_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="topalg_1.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E12:63_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E8:63_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E13:63_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E10:63_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="borsuk_6.html#T83">BORSUK_6:83</a></i>;<br/><a NAME="E14:63_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E1:63"><i><font color="Green">E40</font></i></a>, <a class="ref" href="topalg_1.html#T44" target="_self">Th44</a></i>;<br/><a NAME="E15:63_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="topalg_1.html#E13:63_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="borsuk_6.html#T87">BORSUK_6:87</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E2:63_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topalg_1.html#E4:63_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green">Lemma36</font></i></a></i>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:63"><i><font color="Green">E40</font></i></a>, <a class="ref" href="topalg_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E15:63_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topalg_1.html#T47" target="_self">Th47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_1.html#E6:63_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topalg_1.html#E8:63_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_1.html#E54"><i><font color="Green">Lemma36</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:63"/>
 <a href="topalg_1.html#K4" target="_self">pi_1-iso</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>


</div>
