<?xml version="1.0"?>
<div class="add">

<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>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> ;<br/>









<b>assume </b><a NAME="E1:61"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <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>  )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E2:61"/><i><font color="Green">E49</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>  )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</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/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a>  as    <a href="borsuk_2.html#M1">Path</a> of  <a href="borsuk_1.html#K21">0[01]</a> , <a href="borsuk_1.html#K22">1[01]</a>  <b>by </b><i><a class="ref" href="borsuk_6.html#T12" target="_self">Th12</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  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"><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="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T40">YELLOW12:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> 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"><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="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E6:61"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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 <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:61_1"/><i><font color="Green">E51</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>13</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="ref" href="jordan5b.html#T4">JORDAN5B:4</a></i>;<br/>
<a NAME="E2:61_1"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <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 <a href="topmetr.html#K5">I[01]</a> ,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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:61_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <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> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#D5" target="_self">Def5</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:61_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
;<br/>
<b>hence </b><a NAME="E5:61_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  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"><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="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T39">YELLOW12:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font> 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"><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="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E9:61"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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 <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:61_2"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <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 <a href="topmetr.html#K5">I[01]</a> ,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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E2:61_2"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <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="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="borsuk_1.html#K7">]</a></span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="funct_3.html#K9">pr1</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  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"><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="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T39">YELLOW12:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  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"><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="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T40">YELLOW12:40</a></i>;<br/>
<a NAME="E12:61"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</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> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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_6.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E13:61"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</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> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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_6.html#T8" target="_self">Th8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</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>,<a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E15:61"/><i><font color="Green">E54</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</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><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <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> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E12:61"><i><font color="Green">E52</font></i></a>, <a class="ref" href="borsuk_6.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E16:61"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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 <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:61_3"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:61"><i><font color="Green">E50</font></i></a>, <a class="txt" href="borsuk_6.html#E9:61"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:61_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E15:61"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>17</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>,<a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E18:61"/><i><font color="Green">E56</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</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><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E13:61"><i><font color="Green">E53</font></i></a>, <a class="ref" href="borsuk_6.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E19:61"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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 <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:61_4"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a>, <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E2:61_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E18:61"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:61"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</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> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="pre_topc.html#NM2">Point</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>;<br/>

<a NAME="E1:61_5"/>
<font color="Maroon">c<sub>18</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"><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="E2:61_5"/><b>then </b>
<font color="Maroon">c<sub>18</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 <a href="topmetr.html#K5">I[01]</a> ,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/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:61_5"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>19</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>  &amp; <font color="Maroon">c<sub>20</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>  &amp; <font color="Maroon">c<sub>18</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>19</sub></font>,<font color="Maroon">c<sub>20</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/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</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#E4:61_5"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_5_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>21</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_1_1"><b>suppose </b></a><a NAME="E1:61_5_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><a NAME="E2:61_5_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:61_5_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:61_5_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T174">XREAL_1:174</a>, <a class="ref" href="xreal_1.html#T175">XREAL_1:175</a></i>;<br/><i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E16:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="borsuk_6.html#E4:61_5"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E19:61"><i><font color="Green">E57</font></i></a>, <a class="txt" href="borsuk_6.html#E4:61_5"><i><font color="Green">E58</font></i></a></i>
;<br/><a NAME="E5:61_5_1_1"/><i><font color="Green">E62</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E6:61_5_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E7:61_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="borsuk_6.html#E3:61_5_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:61_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> 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#E3:61_5_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="borsuk_6.html#E4:61_5_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="borsuk_6.html#E5:61_5_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_1_2"><b>suppose </b></a><a NAME="E1:61_5_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>24</sub></font> = 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font>;<br/><a NAME="E2:61_5_1_2"/>
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</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="ref" href="jordan5b.html#T4">JORDAN5B:4</a></i>;<br/><a NAME="E3:61_5_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font> &amp; 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</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="E4:61_5_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:61_5_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T174">XREAL_1:174</a>, <a class="ref" href="xreal_1.html#T175">XREAL_1:175</a></i>;<br/><i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E16:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="borsuk_6.html#E4:61_5"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E19:61"><i><font color="Green">E57</font></i></a>, <a class="txt" href="borsuk_6.html#E4:61_5"><i><font color="Green">E58</font></i></a></i>
;<br/><a NAME="E6:61_5_1_2"/><i><font color="Green">E62</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E7:61_5_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E8:61_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:61_5_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:61_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> 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#E4:61_5_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="borsuk_6.html#E5:61_5_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="borsuk_6.html#E6:61_5_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>18</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>,<a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E22:61"/><i><font color="Green">E58</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</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><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E15:61"><i><font color="Green">E54</font></i></a>, <a class="txt" href="borsuk_6.html#E18:61"><i><font color="Green">E56</font></i></a>, <a class="ref" href="borsuk_6.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E23:61"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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 <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:61_6"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E16:61"><i><font color="Green">E55</font></i></a>, <a class="txt" href="borsuk_6.html#E19:61"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:61_6"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E22:61"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:61"/><i><font color="Green">E60</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 <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,0</span><a href="tarski.html#K4">]</a></span> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>thus </b><font color="Maroon">c<sub>18</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>19</sub></font>,0</span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:61"><i><font color="Green">E49</font></i></a>, <a class="txt" href="borsuk_6.html#E23:61"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E25:61"/><i><font color="Green">E61</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 <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>thus </b><font color="Maroon">c<sub>18</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>19</sub></font>,1</span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:61"><i><font color="Green">E49</font></i></a>, <a class="txt" href="borsuk_6.html#E23:61"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E26:61"/><i><font color="Green">E62</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 <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = 0 as   <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>thus </b><font color="Maroon">c<sub>18</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>19</sub></font></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K3">*</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E23:61"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:61"><i><font color="Green">E48</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E27:61"/><i><font color="Green">E63</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 <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = 1 as   <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>thus </b><font color="Maroon">c<sub>18</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>19</sub></font></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E23:61"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:61"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
1

;<br/>


</div><b>end;</b></div>
<i><font color="Green">E64</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</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"><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="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 <a href="topmetr.html#K5">I[01]</a> ,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/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font>;<br/>
<a NAME="E29:61"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:61"><i><font color="Green">E48</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/>
<a NAME="E30:61"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E22:61"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E31:61"/><i><font color="Green">E66</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/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</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>4</sub></font> <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="61_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:61_12"/>
( <font color="Maroon">c<sub>20</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>  &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></i>;<br/>
<a NAME="E2:61_12"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</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>18</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E28:61"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:61_12"/>
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></i>;<br/>
<a NAME="E4:61_12"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</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>18</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E28:61"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:61_12"/>
<font color="Maroon">c<sub>20</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="E6:61_12"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>,0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,0</span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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>20</sub></font>,0</span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:61_12"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E24:61"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:61_12"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:61"><i><font color="Green">E48</font></i></a>, <a class="ref" href="borsuk_6.html#D6" target="_self">Def6</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>,1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,1</span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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>20</sub></font>,1</span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:61_12"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E25:61"><i><font color="Green">E61</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E32:61"/>
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/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:61_13"/>
( <font color="Maroon">c<sub>20</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>  &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></i>;<br/>
<a NAME="E2:61_13"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">c<sub>20</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>18</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E28:61"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:61_13"/>
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></i>;<br/>
<a NAME="E4:61_13"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>20</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>18</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E28:61"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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>20</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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>20</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:61_13"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="borsuk_6.html#E26:61"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:61"><i><font color="Green">E48</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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>20</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</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>20</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:61_13"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="borsuk_6.html#E27:61"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:61"><i><font color="Green">E48</font></i></a>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E33:61"/>
 <a href="borsuk_6.html#K2" target="_self">RePar</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E30:61"><i><font color="Green">E65</font></i></a>, <a class="txt" href="borsuk_6.html#E31:61"><i><font color="Green">E66</font></i></a>, <a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a></i>;<br/>


</div>
