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<b>consider </b><font color="Maroon">P1</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P2</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E2:7"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">P1</font> <a href="topreal1.html#NR3" target="_self">is_S-P_arc</a>  &amp; <font color="Maroon">P2</font> <a href="topreal1.html#NR3" target="_self">is_S-P_arc</a>  &amp;  <a href="topreal1.html#K2">R^2-unit_square</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#T34">TOPREAL1:34</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span><b> such that </b><br/><a NAME="E4:7"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, <a class="ref" href="topreal1.html#T36">TOPREAL1:36</a></i>;<br/>
<a NAME="E5:7"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">f0</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span><b> such that </b><br/><a NAME="E7:7"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f0</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, <a class="ref" href="topreal1.html#T36">TOPREAL1:36</a></i>;<br/>
<a NAME="E8:7"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">f0</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E9:7"/><i><font color="Green">E62</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E10:7"/><i><font color="Green">E63</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E12:7"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i>, <a class="txt" href="topreal2.html#E4"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="topreal2.html#E5"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="compts_1.html#T23">COMPTS_1:23</a></i>;<br/>
<a NAME="E13:7"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">P1</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/>
<a NAME="E14:7"/><i><font color="Green">E65</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f0</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P2</font> = <font color="Maroon">P2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E16:7"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i>, <a class="txt" href="topreal2.html#E4"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="topreal2.html#E7"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="compts_1.html#T23">COMPTS_1:23</a></i>;<br/>
<a NAME="E17:7"/><b>then </b>
<font color="Maroon">P2</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/>
<b>hence </b><a NAME="E18:7"/>
 <a href="topreal1.html#K2">R^2-unit_square</a>  is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i>, <a class="txt" href="topreal2.html#E6"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="compts_1.html#T19">COMPTS_1:19</a></i>;<br/>


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