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<b>consider </b><font color="Maroon" title="c10">P1</font>, <font color="Maroon" title="c11">P2</font> being  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><b> such that </b><br/><a NAME="E2:7"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c10">P1</font> <a href="topreal1.html#NR3" title="TOPREAL1:NR.3">is_S-P_arc</a>  &amp; <font color="Maroon" title="c11">P2</font> <a href="topreal1.html#NR3" title="TOPREAL1:NR.3">is_S-P_arc</a>  &amp;  <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">P2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#T34" title="TOPREAL1:th.34">TOPREAL1:34</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</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">P1</font></span>)</span><b> such that </b><br/><a NAME="E4:7"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topreal2.html#E2:7"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="topreal1.html#T36" title="TOPREAL1:th.36">TOPREAL1:36</a></i>;<br/>
<a NAME="E5:7"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c12">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E4:7"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">f0</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</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="c11">P2</font></span>)</span><b> such that </b><br/><a NAME="E7:7"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c13">f0</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topreal2.html#E2:7"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="topreal1.html#T36" title="TOPREAL1:th.36">TOPREAL1:36</a></i>;<br/>
<a NAME="E8:7"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c13">f0</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E7:7"><i><font color="Green" title="E32">A4</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="E9:7"/><i><font color="Green" title="E34">A6</font></i>: 
 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11" title="HEINE:th.11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E10:7"/><i><font color="Green" title="E35">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">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">P1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E4:7"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">P1</font> = <font color="Maroon" title="c10">P1</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="E12:7"/>
<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="c14">P1</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#E5:7"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="topreal2.html#E9:7"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="topreal2.html#E10:7"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="compts_1.html#T23" title="COMPTS_1:th.23">COMPTS_1:23</a></i>;<br/>
<a NAME="E13:7"/><b>then </b><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c14">P1</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>;<br/>
<a NAME="E14:7"/><i><font color="Green" title="E37">A9</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">f0</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="c11">P2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E7:7"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">P2</font> = <font color="Maroon" title="c11">P2</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="E16:7"/>
<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="c15">P2</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#E8:7"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="topreal2.html#E9:7"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="topreal2.html#E14:7"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="compts_1.html#T23" title="COMPTS_1:th.23">COMPTS_1:23</a></i>;<br/>
<a NAME="E17:7"/><b>then </b>
<font color="Maroon" title="c15">P2</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>;<br/>
<b>hence </b><a NAME="E18:7"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 <b>by </b><i><a class="txt" href="topreal2.html#E2:7"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="topreal2.html#E13:7"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="compts_1.html#T19" title="COMPTS_1:th.19">COMPTS_1:19</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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