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<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K7">]</a></span>,<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="borsuk_1.html#K7">]</a></span>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="borsuk_6.html#M1">Homotopy</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>;<br/>























<b>assume </b><a NAME="E1:33"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a>  <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R2">=</a>  <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R2">=</a>  <a href="topalg_4.html#K5" target="_self">pr2</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#R2">=</a>  <a href="topalg_4.html#K5" target="_self">pr2</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a>  &amp; <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:33"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_6.html#D13">BORSUK_6:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>take </b>
<span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a><br/>

<b>thus </b><a NAME="E1:33_1"/>
( <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> 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">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">K537</a>()) holds <br/>( <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <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>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <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>11</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="subset_1.html#M1">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">K537</a>()) holds <br/>( <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="borsuk_1.html#K7">]</a></span> ) ) ) ) )
 <b>by </b><i><a class="txt" href="topalg_4.html#E1:33"><i><font color="Green">E24</font></i></a>, <a class="txt" href="topalg_4.html#E27"><i><font color="Green">Lemma23</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E3:33"/>
( <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> 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">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">K537</a>()) holds <br/>( <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <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>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <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>11</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="subset_1.html#M1">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">K537</a>()) holds <br/>( <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <span class="p1"><a href="topalg_4.html#K3" target="_self">&lt;:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="topalg_4.html#K3" target="_self">:&gt;</a></span> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="borsuk_1.html#K7">]</a></span> ) ) ) ) )
 <b>by </b><i><a class="txt" href="topalg_4.html#E1:33"><i><font color="Green">E24</font></i></a>, <a class="txt" href="topalg_4.html#E27"><i><font color="Green">Lemma23</font></i></a></i>;<br/>


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