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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:79"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:79"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:79"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = 0, <font color="Maroon">c<sub>7</sub></font> = 1 <a href="xcmplx_0.html#K7">/</a> 2, <font color="Maroon">c<sub>8</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/>
<a NAME="E5:79"/><i><font color="Green">E72</font></i>: 
( 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>  &amp; 1 <a href="xcmplx_0.html#K7">/</a> 2 <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/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a> , <font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:79"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="rcomp_2.html#K2">.]</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:79"><i><font color="Green">E69</font></i></a>, <a class="ref" href="borsuk_4.html#T74" target="_self">Th74</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a> , <font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E9:79"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E2:79"><i><font color="Green">E70</font></i></a>, <a class="ref" href="borsuk_4.html#T75" target="_self">Th75</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="borsuk_4.html#K1" target="_self">I(01)</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<a NAME="E10:79"/>
<font color="Maroon">c<sub>3</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:79"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan5b.html#T14">JORDAN5B:14</a></i>;<br/>
<a NAME="E11:79"/><b>then </b><i><font color="Green">E75</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>3</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E1:79"><i><font color="Green">E69</font></i></a>, <a class="ref" href="borsuk_4.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E12:79"/>
<font color="Maroon">c<sub>2</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E13:79"/><b>then </b>
not <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E11:79"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E14:79"/><b>then </b>
not 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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> 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> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E16:79"/><i><font color="Green">E76</font></i>: 
not 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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E11:79"><i><font color="Green">E75</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E17:79"/>
not 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> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E11:79"><i><font color="Green">E75</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>, <font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> 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> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="borsuk_4.html#E16:79"><i><font color="Green">E76</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E19:79"/><i><font color="Green">E77</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E20:79"/><i><font color="Green">E78</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E21:79"/><i><font color="Green">E79</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E22:79"/><i><font color="Green">E80</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E23:79"/><b>then </b><i><font color="Green">E81</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E21:79"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E24:79"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<a NAME="E25:79"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E26:79"/><b>then </b><i><font color="Green">E83</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E22:79"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E27:79"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="txt" href="borsuk_4.html#E23:79"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E29:79"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E30:79"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E24:79"><i><font color="Green">E82</font></i></a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="txt" href="borsuk_4.html#E26:79"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E32:79"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E33:79"/><b>then </b><i><font color="Green">E86</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="borsuk_4.html#E27:79"><i><font color="Green">E84</font></i></a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>, <font color="Maroon">c<sub>25</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a>  <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E35:79"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="rcomp_2.html#K2">.]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E36:79"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>24</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E5:79"><i><font color="Green">E72</font></i></a>, <a class="ref" href="borsuk_4.html#T72" target="_self">Th72</a></i>;<br/>
<a NAME="E37:79"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>,1</span><a href="rcomp_2.html#K1">.[</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E38:79"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>25</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E5:79"><i><font color="Green">E72</font></i></a>, <a class="ref" href="borsuk_4.html#T73" target="_self">Th73</a></i>;<br/>
<i><font color="Green">E91</font></i>: <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</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"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E35:79"><i><font color="Green">E87</font></i></a>, <a class="txt" href="borsuk_4.html#E37:79"><i><font color="Green">E89</font></i></a>, <a class="ref" href="borsuk_4.html#T40" target="_self">Th40</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a> 
<b>by </b><i><a class="ref" href="borsuk_4.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_4.html#K1" target="_self">I(01)</a> 

;<br/>
<a NAME="E40:79"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</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"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E35:79"><i><font color="Green">E87</font></i></a>, <a class="txt" href="borsuk_4.html#E37:79"><i><font color="Green">E89</font></i></a>, <a class="ref" href="borsuk_4.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E41:79"/>
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"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</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"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>10</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>12</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="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:79_2"/>
<font color="Maroon">c<sub>26</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"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</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"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:79_2"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E40:79"><i><font color="Green">E92</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:79_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a> ,<font color="Maroon">c<sub>19</sub></font><b> such that </b><br/><a NAME="E43:79"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>12</sub></font> &amp; <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="borsuk_4.html#E30:79"><i><font color="Green">E85</font></i></a>, <a class="txt" href="borsuk_4.html#E33:79"><i><font color="Green">E86</font></i></a>, <a class="txt" href="borsuk_4.html#E36:79"><i><font color="Green">E88</font></i></a>, <a class="txt" href="borsuk_4.html#E38:79"><i><font color="Green">E90</font></i></a>, <a class="txt" href="borsuk_4.html#E39:79"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jgraph_2.html#T9">JGRAPH_2:9</a></i>;<br/>
<i><font color="Green">E94</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_4.html#K1" target="_self">I(01)</a> 

;<br/>
<a NAME="E45:79"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E46:79"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>23</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E47:79"/><i><font color="Green">E97</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E48:79"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E35:79"><i><font color="Green">E87</font></i></a>, <a class="txt" href="borsuk_4.html#E37:79"><i><font color="Green">E89</font></i></a>, <a class="ref" href="borsuk_4.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E49:79"/><i><font color="Green">E99</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</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="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E50:79"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E51:79"/><i><font color="Green">E101</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</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="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E52:79"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then </b><i><font color="Green">E103</font></i>: <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E50:79"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T41">XBOOLE_1:41</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:79"><i><font color="Green">E71</font></i></a>, <a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="zfmisc_1.html#T23">ZFMISC_1:23</a></i>
;<br/>
<a NAME="E54:79"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>26</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E43:79"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_4.html#E45:79"><i><font color="Green">E95</font></i></a>, <a class="txt" href="borsuk_4.html#E46:79"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_4.html#E48:79"><i><font color="Green">E98</font></i></a>, <a class="ref" href="borsuk_4.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E55:79"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E45:79"><i><font color="Green">E95</font></i></a>, <a class="txt" href="borsuk_4.html#E49:79"><i><font color="Green">E99</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E56:79"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E46:79"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_4.html#E51:79"><i><font color="Green">E101</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E57:79"/>
<font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">c<sub>23</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_5"><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a><br/>

<b>assume </b><a NAME="E1:79_5"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:79_5"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E48:79"><i><font color="Green">E98</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:79_5"/>
<font color="Maroon">c<sub>22</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>23</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="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E107</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>26</sub></font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E43:79"><i><font color="Green">E93</font></i></a>, <a class="ref" href="frechet.html#T39">FRECHET:39</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="borsuk_4.html#E50:79"><i><font color="Green">E100</font></i></a>, <a class="txt" href="borsuk_4.html#E52:79"><i><font color="Green">E102</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E59:79"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E54:79"><i><font color="Green">E104</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E60:79"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a>, <a class="txt" href="borsuk_4.html#E43:79"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_4.html#E45:79"><i><font color="Green">E95</font></i></a>, <a class="txt" href="borsuk_4.html#E46:79"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_4.html#E48:79"><i><font color="Green">E98</font></i></a>, <a class="txt" href="borsuk_4.html#E53:79"><i><font color="Green">E103</font></i></a>, <a class="txt" href="borsuk_4.html#E55:79"><i><font color="Green">E105</font></i></a>, <a class="txt" href="borsuk_4.html#E56:79"><i><font color="Green">E106</font></i></a>, <a class="ref" href="borsuk_4.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E61:79"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>20</sub></font>,<a href="borsuk_4.html#K1" target="_self">I(01)</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E63:79"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_4.html#K1" target="_self">I(01)</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>21</sub></font>,<a href="borsuk_4.html#K1" target="_self">I(01)</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E65:79"/><i><font color="Green">E109</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E19:79"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E66:79"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E20:79"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>30</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="txt" href="borsuk_4.html#E65:79"><i><font color="Green">E109</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E68:79"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E19:79"><i><font color="Green">E77</font></i></a>, <a class="txt" href="borsuk_4.html#E20:79"><i><font color="Green">E78</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E69:79"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>29</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_7"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E2:79_7"/>
<font color="Maroon">c<sub>31</sub></font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E1:79"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/>
<a NAME="E3:79_7"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>31</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T16">COMPTS_1:16</a></i>;<br/>
<a NAME="E4:79_7"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E5:79_7"/><i><font color="Green">E114</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:79"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:79_7"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E1:79"><i><font color="Green">E69</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:79_7"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:79_7"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E5:79_7"><i><font color="Green">E114</font></i></a>, <a class="ref" href="zfmisc_1.html#T40">ZFMISC_1:40</a></i>;<br/>
<i><font color="Green">E116</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:79"><i><font color="Green">E71</font></i></a>, <a class="ref" href="zfmisc_1.html#T23">ZFMISC_1:23</a></i>
;<br/>
<font color="Maroon">c<sub>31</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<font color="Maroon">c<sub>31</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E4:79_7"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E8:79_7"><i><font color="Green">E115</font></i></a>, <a class="txt" href="borsuk_4.html#E9:79_7"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E11:79_7"/>
<font color="Maroon">c<sub>29</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:79_7"><i><font color="Green">E112</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E70:79"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>30</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_8"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E2:79_8"/>
<font color="Maroon">c<sub>31</sub></font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:79"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/>
<a NAME="E3:79_8"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>31</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T16">COMPTS_1:16</a></i>;<br/>
<a NAME="E4:79_8"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E5:79_8"/><i><font color="Green">E115</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:79"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:79_8"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:79"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:79_8"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:79_8"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E5:79_8"><i><font color="Green">E115</font></i></a>, <a class="ref" href="zfmisc_1.html#T40">ZFMISC_1:40</a></i>;<br/>
<i><font color="Green">E117</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E3:79"><i><font color="Green">E71</font></i></a>, <a class="ref" href="zfmisc_1.html#T23">ZFMISC_1:23</a></i>
;<br/>
<font color="Maroon">c<sub>31</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
<font color="Maroon">c<sub>31</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E4:79_8"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <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>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="borsuk_4.html#E8:79_8"><i><font color="Green">E116</font></i></a>, <a class="txt" href="borsuk_4.html#E9:79_8"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E11:79_8"/>
<font color="Maroon">c<sub>30</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:79_8"><i><font color="Green">E113</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E71:79"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E72:79"/><b>then </b><i><font color="Green">E113</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="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E73:79"/>
<font color="Maroon">c<sub>12</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E74:79"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>28</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E75:79"/>
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> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>27</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>28</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="79_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>31</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:79_9"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:79_9"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="txt" href="borsuk_4.html#E49:79"><i><font color="Green">E99</font></i></a>, <a class="txt" href="borsuk_4.html#E51:79"><i><font color="Green">E101</font></i></a>, <a class="txt" href="borsuk_4.html#E53:79"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:79_9"/>
<font color="Maroon">c<sub>7</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="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E4:79_9"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <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="txt" href="borsuk_4.html#E47:79"><i><font color="Green">E97</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E5:79_9"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E6:79_9"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <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="txt" href="borsuk_4.html#E47:79"><i><font color="Green">E97</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E7:79_9"/>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E7:79"><i><font color="Green">E73</font></i></a>, <a class="txt" href="borsuk_4.html#E45:79"><i><font color="Green">E95</font></i></a>, <a class="txt" href="borsuk_4.html#E55:79"><i><font color="Green">E105</font></i></a>, <a class="txt" href="borsuk_4.html#E4:79_9"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>
<b>hence </b><a NAME="E8:79_9"/>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E9:79"><i><font color="Green">E74</font></i></a>, <a class="txt" href="borsuk_4.html#E46:79"><i><font color="Green">E96</font></i></a>, <a class="txt" href="borsuk_4.html#E56:79"><i><font color="Green">E106</font></i></a>, <a class="txt" href="borsuk_4.html#E2:79_9"><i><font color="Green">E115</font></i></a>, <a class="txt" href="borsuk_4.html#E6:79_9"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>19</sub></font>,<a href="borsuk_4.html#K1" target="_self">I(01)</a> <b> such that </b><br/><a NAME="E77:79"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E24:79"><i><font color="Green">E82</font></i></a>, <a class="txt" href="borsuk_4.html#E27:79"><i><font color="Green">E84</font></i></a>, <a class="txt" href="borsuk_4.html#E68:79"><i><font color="Green">E110</font></i></a>, <a class="txt" href="borsuk_4.html#E69:79"><i><font color="Green">E111</font></i></a>, <a class="txt" href="borsuk_4.html#E70:79"><i><font color="Green">E112</font></i></a>, <a class="txt" href="borsuk_4.html#E72:79"><i><font color="Green">E113</font></i></a>, <a class="txt" href="borsuk_4.html#E74:79"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jgraph_2.html#T9">JGRAPH_2:9</a></i>;<br/>
<a NAME="E78:79"/>
<font color="Maroon">c<sub>26</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E43:79"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_4.html#E44:79"><i><font color="Green">E94</font></i></a>, <a class="txt" href="borsuk_4.html#E54:79"><i><font color="Green">E104</font></i></a>, <a class="txt" href="borsuk_4.html#E58:79"><i><font color="Green">E107</font></i></a>, <a class="txt" href="borsuk_4.html#E60:79"><i><font color="Green">E108</font></i></a>, <a class="txt" href="borsuk_4.html#E77:79"><i><font color="Green">E115</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E79:79"/>
 <a href="borsuk_4.html#K1" target="_self">I(01)</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="t_0topsp.html#R1">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a></i>;<br/>


</div>
