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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#V6" title="COMPTS_1:attr.6">compact</a> ) )</span><br/>

<b>given </b><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#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="c10">P</font></span>)</span><b> such that </b><a NAME="E2:21_1_1"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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#V6" title="COMPTS_1:attr.6">compact</a> )</span><br/>

<a NAME="E3:21_1_1"/><i><font color="Green" title="E38">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>  is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="ref" href="topreal2.html#T2" target="_self" title="TOPREAL2:th.2">Th2</a>, <a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></i>;<br/>
<b>thus </b><a NAME="E4:21_1_1"/>
not <font color="Maroon" title="c10">P</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_1"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c10">P</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <font color="Maroon" title="c10">P</font></span>)</span> = 
 <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>

<br/>.= 
<font color="Maroon" title="c10">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E3:21_1_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="c10">P</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E1:21_1_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal2.html#E2:21_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="topreal2.html#E37"><i><font color="Green" title="E36">Lm30</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>

<b>then reconsider </b><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> ;<br/>
<a NAME="E6:21_1_1"/>
( <font color="Maroon" title="c11">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E2:21_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:21_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="c12">R</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E3:21_1_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="compts_1.html#T23" title="COMPTS_1:th.23">COMPTS_1:23</a></i>;<br/>
<b>hence </b><a NAME="E8:21_1_1"/>
<font color="Maroon" title="c10">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</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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