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<b>let </b><font color="Maroon" title="c9">S</font>, <font color="Maroon" title="c10">T</font> be  <a href="toprealb.html#V1" title="TOPREALB:attr.1">being_simple_closed_curve</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">S</font>,<font color="Maroon" title="c10">T</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> </span><br/>

<a NAME="E1:42"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c9">S</font> #), <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c10">T</font> #) <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">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" title="c11">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c9">S</font> #) as   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a> <b>by </b><i><a class="ref" href="toprealb.html#D5" target="_self" title="TOPREALB:def.5">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">B</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c10">T</font> #) as   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a> <b>by </b><i><a class="ref" href="toprealb.html#D5" target="_self" title="TOPREALB:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">A</font></span>)</span><b> such that </b><br/><a NAME="E4:42_1"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="ref" href="topreal2.html#D1" title="TOPREAL2:def.1">TOPREAL2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">B</font></span>)</span><b> such that </b><br/><a NAME="E6:42_1"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="ref" href="topreal2.html#D1" title="TOPREAL2:def.1">TOPREAL2:def 1</a></i>;<br/>
<a NAME="E7:42_1"/><i><font color="Green" title="E25">A3</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c9">S</font> #) is  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tmap_1.html#T11" title="TMAP_1:th.11">TMAP_1:11</a></i>;<br/>
<a NAME="E8:42_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c9">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A</font>
 
;<br/>
<a NAME="E9:42_1"/><b>then </b><i><font color="Green" title="E26">A4</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c9">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">A</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:42_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E10:42_1"/><i><font color="Green" title="E27">A5</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c10">T</font> #) is  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tmap_1.html#T11" title="TMAP_1:th.11">TMAP_1:11</a></i>;<br/>
<a NAME="E11:42_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c10">T</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B</font>
 
;<br/>
<a NAME="E12:42_1"/><b>then </b><i><font color="Green" title="E28">A6</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c10">T</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">B</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E10:42_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">h</font> = <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c9">S</font> #),<a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c10">T</font> #) <b>by </b><i><a class="txt" href="toprealb.html#E9:42_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">h</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">h</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a></span><br/>

<a NAME="E14:42_1"/>
<font color="Maroon" title="c13">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:42_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="tops_2.html#T70" title="TOPS_2:th.70">TOPS_2:70</a></i>;<br/>
<b>hence </b><a NAME="E15:42_1"/>
<font color="Maroon" title="c15">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E6:42_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="toprealb.html#E9:42_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="toprealb.html#E12:42_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="tops_2.html#T71" title="TOPS_2:th.71">TOPS_2:71</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:42"/>
<font color="Maroon" title="c9">S</font>,<font color="Maroon" title="c10">T</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="topreala.html#T36" title="TOPREALA:th.36">TOPREALA:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
