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<b>let </b><font color="Maroon" title="c9">S</font> be    <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> is <a href="toprealb.html#V1" title="TOPREALB:attr.1">being_simple_closed_curve</a> implies ( not <font color="Maroon" title="c9">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c9">S</font> is <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a> &amp; <font color="Maroon" title="c9">S</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> ) )</span><br/>

<b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S</font> is   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="toprealb.html#D5" target="_self" title="TOPREALB:def.5">TOPREALB:def 5</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="c9">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c9">S</font> is <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a> &amp; <font color="Maroon" title="c9">S</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c10">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S</font> as   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a> ;<br/>
<a NAME="E3:33_1_1"/>
not <font color="Maroon" title="c10">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<b>hence </b><a NAME="E4:33_1_1"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0: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="c9">S</font> is <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a> &amp; <font color="Maroon" title="c9">S</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )</span><br/>

<b>thus </b><a NAME="E5:33_1_1"/>
<font color="Maroon" title="c9">S</font> is <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:33_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="topalg_3.html#T11" title="TOPALG_3:th.11">TOPALG_3:11</a></i>; <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> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><br/>

<a NAME="E6:33_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c9">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A</font>
 
;<br/>
<a NAME="E7:33_1_1"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c9">S</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#T11" title="COMPTS_1:th.11">COMPTS_1:11</a></i>;<br/>
<b>hence </b><a NAME="E8:33_1_1"/>
<font color="Maroon" title="c9">S</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T10" title="COMPTS_1:th.10">COMPTS_1:10</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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