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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>given </b><font color="Maroon">c<sub>2</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>1</sub></font></span>)</span><b> such that </b><a NAME="E1:42_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal2.html#D1">TOPREAL2:def 1</a><br/>

<a NAME="E2:42_1_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <a href="topreal1.html#K2">R^2-unit_square</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>1</sub></font> <a href="t_0topsp.html#R1">are_homeomorphic</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a><br/>

<b>thus </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topreal6.html#E1:42_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:42_1_1"/>
<span class="p1">(<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>  is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<a NAME="E4:42_1_1"/><b>then </b>
<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>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i><a class="txt" href="topreal6.html#E2:42_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="topreal6.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:42_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>


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