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<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be  <a href="toprealb.html#V1" target="_self">being_simple_closed_curve</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2;<br/>

<a NAME="E1:42"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font> #), <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>10</sub></font> #) <a href="t_0topsp.html#R1">are_homeomorphic</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font> #) as   <a href="topreal2.html#NM1">Simple_closed_curve</a> <b>by </b><i><a class="ref" href="toprealb.html#D5" target="_self">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>10</sub></font> #) as   <a href="topreal2.html#NM1">Simple_closed_curve</a> <b>by </b><i><a class="ref" href="toprealb.html#D5" target="_self">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM7">Function</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> <a href="topreal1.html#K2">R^2-unit_square</a> </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>11</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:42_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="ref" href="topreal2.html#D1">TOPREAL2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM7">Function</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> <a href="topreal1.html#K2">R^2-unit_square</a> </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>12</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:42_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="ref" href="topreal2.html#D1">TOPREAL2:def 1</a></i>;<br/>
<a NAME="E7:42_1"/><i><font color="Green">E25</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font> #) is  <a href="pre_topc.html#V1">strict</a>  <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="tmap_1.html#T11">TMAP_1:11</a></i>;<br/>
<a NAME="E8:42_1"/>
 <a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
;<br/>
<a NAME="E9:42_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font> #) <a href="hidden.html#R1">=</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> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:42_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E10:42_1"/><i><font color="Green">E27</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>10</sub></font> #) is  <a href="pre_topc.html#V1">strict</a>  <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="tmap_1.html#T11">TMAP_1:11</a></i>;<br/>
<a NAME="E11:42_1"/>
 <a href="pre_topc.html#K2">[#]</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>10</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
;<br/>
<a NAME="E12:42_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>10</sub></font> #) <a href="hidden.html#R1">=</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> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E10:42_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> as   <a href="struct_0.html#NM7">Function</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>9</sub></font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>10</sub></font> #) <b>by </b><i><a class="txt" href="toprealb.html#E9:42_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a><br/>

<a NAME="E14:42_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:42_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tops_2.html#T70">TOPS_2:70</a></i>;<br/>
<b>hence </b><a NAME="E15:42_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E6:42_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="toprealb.html#E9:42_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="toprealb.html#E12:42_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:42"/>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="borsuk_3.html#R1">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="topreala.html#T36">TOPREALA:36</a></i>;<br/>


</div>
