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

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>













<b>assume </b><a NAME="E1:107"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>14</sub></font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:107"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E3:107"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:107"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E5:107"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E6:107"/><i><font color="Green">E86</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E7:107"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E8:107"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E9:107"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E10:107"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E11:107"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E12:107"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E12:107"><i><font color="Green">E92</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2 ;<br/>
<a NAME="E15:107"/><i><font color="Green">E93</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E16:107"/><i><font color="Green">E94</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E17:107"/><i><font color="Green">E95</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3">BROUWER:3</a></i>;<br/>
<a NAME="E18:107"/><i><font color="Green">E96</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T16">TOPREAL9:16</a></i>;<br/>
<a NAME="E19:107"/><i><font color="Green">E97</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E15:107"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jordan.html#E4:107"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E20:107"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/>
<a NAME="E21:107"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E17:107"><i><font color="Green">E95</font></i></a>, <a class="ref" href="topreal9.html#T8">TOPREAL9:8</a></i>;<br/>
<a NAME="E22:107"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E23:107"/><i><font color="Green">E100</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan.html#E6:107"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E17:107"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan.html#E18:107"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan.html#E23:107"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan.html#E22:107"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:107_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan.html#E10:107"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan.html#E11:107"><i><font color="Green">E91</font></i></a>, <a class="ref" href="connsp_1.html#T37">CONNSP_1:37</a></i>;<br/><b>hence </b><a NAME="E3:107_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E4:107"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan.html#E9:107"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan.html#E1:107_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E26:107"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E21:107"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font>;<br/>
<a NAME="E28:107"/><i><font color="Green">E101</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E29:107"/><i><font color="Green">E102</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E30:107"/><i><font color="Green">E103</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E31:107"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:107_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:107_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E2:107"><i><font color="Green">E82</font></i></a></i>;<br/>
<a NAME="E3:107_2"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E3:107"><i><font color="Green">E83</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:107_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E12:107"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan.html#E19:107"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan.html#E2:107_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan.html#E18:107"><i><font color="Green">E96</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:107"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:107_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<a NAME="E2:107_3"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_3_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:107_3_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:107"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan.html#E9:107"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan.html#E10:107"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan.html#E11:107"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jordan.html#T68" target="_self">Th68</a></i>;<br/><b>hence </b><a NAME="E3:107_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E4:107"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan.html#E1:107_3_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:107_3"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E2:107_3"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan.html#E1:107_3"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan.html#E17:107"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan.html#E12:107"><i><font color="Green">E92</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font> <b>by </b><i><a class="txt" href="jordan.html#E28:107"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan.html#E30:107"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan.html#E32:107"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E34:107"/>
<font color="Maroon">c<sub>16</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E16:107"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>20</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<font color="Maroon">c<sub>18</sub></font> <b>by </b><i><a class="txt" href="jordan.html#E29:107"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan.html#E30:107"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan.html#E31:107"><i><font color="Green">E104</font></i></a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E36:107"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E37:107"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:107_4"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E36:107"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:107_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_4_1"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or  not <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_4_1_1"><b>suppose </b></a><a NAME="E1:107_4_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:107_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E22:107"><i><font color="Green">E99</font></i></a>, <a class="txt" href="jordan.html#E2:107_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:107_4_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E36:107"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_4_1_2"><b>suppose </b></a><a NAME="E1:107_4_1_2"/>
not <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:107_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan.html#E2:107_4"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan.html#E17:107"><i><font color="Green">E95</font></i></a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><a NAME="E3:107_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E2:107_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:107_4_1_2"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E36:107"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> as  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#D14">BORSUK_1:def 14</a>, <a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E39:107"/>
<font color="Maroon">c<sub>25</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E40:107"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T8">TSEP_1:8</a></i>;<br/>
<a NAME="E41:107"/>
<font color="Maroon">c<sub>9</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_5"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a><br/>

<b>thus </b><a NAME="E1:107_5"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="jordan.html#E9:107"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan.html#E10:107"><i><font color="Green">E90</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E42:107"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T18">SPRECT_3:18</a></i>;<br/>
<a NAME="E43:107"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E44:107"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E45:107"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T8">TSEP_1:8</a></i>;<br/>
<a NAME="E46:107"/><i><font color="Green">E110</font></i>: 
 <a href="topgrp_1.html#K3">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E36:107"><i><font color="Green">E106</font></i></a></i>;<br/>
<a NAME="E47:107"/>
<span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T7">TSEP_1:7</a></i>;<br/>
<a NAME="E48:107"/><b>then </b>
<span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E28:107"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan.html#E30:107"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan.html#E32:107"><i><font color="Green">E105</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<a NAME="E49:107"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E46:107"><i><font color="Green">E110</font></i></a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E50:107"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>13</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E16:107"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jordan.html#E30:107"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan.html#E31:107"><i><font color="Green">E104</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E29:107"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E52:107"/>
<font color="Maroon">c<sub>26</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E7:107"><i><font color="Green">E87</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E53:107"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>24</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E50:107"><i><font color="Green">E112</font></i></a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E54:107"/><i><font color="Green">E114</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a>  holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:107_6"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a>  )
 ;<br/>

<a NAME="E2:107_6"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:107_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E5:107"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E4:107_6"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:107"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan.html#E2:107_6"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan.html#E5:107"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:107"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:107_7"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:107_7"/><b>then </b><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:107_7"/><b>then </b>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="txt" href="jordan.html#E36:107"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:107_7"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E54:107"><i><font color="Green">E114</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="jordan.html#E36:107"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan.html#E2:107_7"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="jordan.html#E36:107"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan.html#E2:107_7"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan.html#E4:107_7"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span><b> such that </b><br/><a NAME="E57:107"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E58:107"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan.html#E36:107"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan.html#E37:107"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan.html#E40:107"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan.html#E45:107"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan.html#E49:107"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan.html#E53:107"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jgraph_2.html#T9">JGRAPH_2:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>27</sub></font>
;<br/>

<b>thus </b><a NAME="E59:107"/>
<font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan.html#E58:107"><i><font color="Green">E116</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>28</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>;<br/>

<a NAME="E60:107"/><i><font color="Green">E117</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:107_8"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:107_8"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_8_1"/>
(  not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font> or <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_8_1_1"><b>suppose </b></a><a NAME="E1:107_8_1_1"/>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="jordan.html#E57:107"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="jordan.html#E2:107_8"><i><font color="Green">E119</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_8_1_2"><b>suppose </b></a><a NAME="E1:107_8_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><div class="add"><a NAME="E2:107_8_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i><a class="txt" href="jordan.html#E1:107_8"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan.html#E29:107"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="jordan.html#E1:107_8_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan.html#E57:107"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="jordan.html#E1:107_8_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan.html#E29:107"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="jordan.html#E54:107"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan.html#E2:107_8_1_2"><i><font color="Green">E121</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E62:107"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E63:107"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="jordan.html#E60:107"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan.html#E63:107"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan.html#E29:107"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan.html#E57:107"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="jordan.html#E63:107"><i><font color="Green">E118</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


</div>
