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<b>let </b><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> is <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> implies <font color="Maroon" title="c1">P</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )</span><br/>

<b>given </b><font color="Maroon" title="c2">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="c1">P</font></span>)</span><b> such that </b><a NAME="E2:42_1_1"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal2.html#D1" title="TOPREAL2:def.1">TOPREAL2: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="c1">P</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/>

<a NAME="E3:42_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
<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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</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="c1">P</font> <a href="t_0topsp.html#R1" title="T_0TOPSP:pred.1">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" title="c2">f</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="c2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a></span><br/>

<b>thus </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon" title="c2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>by </b><i><a class="txt" href="#E2:42_1_1"><i><font color="Green" title="E20">A1</font></i></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>
<a NAME="E4:42_1_1"/>
<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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a></i>;<br/>
<a NAME="E5:42_1_1"/><b>then </b>
<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="c1">P</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 
<b>by </b><i><a class="txt" href="#E3:42_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="topreal6.html#T25" target="_self" title="TOPREAL6:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E6:42_1_1"/>
<font color="Maroon" title="c1">P</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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