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<span class="kw">let </span><font color="Maroon" title="c10">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  implies (  not <font color="Maroon" title="c10">P</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c10">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  ) )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c11">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#K1" title="TOPREAL1:func.1">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="c10">P</font></span>)</span><span class="kw"> such that </span><a NAME="E2:20_1_1"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c11">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal2.html#D1" target="_self" title="TOPREAL2:def.1">TOPREAL2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">P</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c10">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:20_1_1"/>
 not <font color="Maroon" title="c10">P</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="topreal2.html#E44"><span class="lab"><font color="Green" title="E43">Lm37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/>

<a NAME="E4:20_1_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <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="c10">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">R</font> = <font color="Maroon" title="c10">P</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="topreal2.html#E44"><span class="lab"><font color="Green" title="E43">Lm37</font></span></a></span>;<br/>
<a NAME="E6:20_1_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c11">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E7:20_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#K1" title="TOPREAL1:func.1">R^2-unit_square</a> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal2.html#T2" target="_self" title="TOPREAL2:th.2">Th2</a>, <a class="ref" href="compts_1.html#T3" title="COMPTS_1:th.3">COMPTS_1:3</a></span>;<br/>
<a NAME="E8:20_1_1"/><span class="kw">then </span>
<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">R</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="compts_1.html#T14" title="COMPTS_1:th.14">COMPTS_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:20_1_1"/>
<font color="Maroon" title="c10">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compts_1.html#T3" title="COMPTS_1:th.3">COMPTS_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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