<?xml version="1.0"?>
<div class="add">

<a NAME="E1:7"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</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="heine.html#T11" title="HEINE:th.11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:7"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c10">P1</font> is  <a href="topreal1.html#V5" title="TOPREAL1:attr.5">being_S-P_arc</a> 
 <span class="kw">and </span><br/><a NAME="E4:7"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c11">P2</font> is  <a href="topreal1.html#V5" title="TOPREAL1:attr.5">being_S-P_arc</a> 
 <span class="kw">and </span><br/><a NAME="E5:7"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="topreal1.html#K1" title="TOPREAL1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#T34" title="TOPREAL1:th.34">TOPREAL1:34</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:7"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c12">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="topreal1.html#T36" title="TOPREAL1:th.36">TOPREAL1:36</a></span>;<br/>
<a NAME="E8:7"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E40">A5</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">consider </span><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><span class="kw"> such that </span><br/><a NAME="E10:7"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<font color="Maroon" title="c13">f0</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="topreal1.html#T36" title="TOPREAL1:th.36">TOPREAL1:36</a></span>;<br/>
<a NAME="E11:7"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:7"><span class="lab"><font color="Green" title="E42">A7</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="c14">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="E13:7"/>
<font color="Maroon" title="c13">f0</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="#E10:7"><span class="lab"><font color="Green" title="E42">A7</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="E14:7"/><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="c14">P2</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="#E1:7"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E11:7"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="compts_1.html#T23" title="COMPTS_1:th.23">COMPTS_1:23</a></span>;<br/>
<a NAME="E15:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon" title="c14">P2</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#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">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="E17:7"/>
<font color="Maroon" title="c12">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="#E7:7"><span class="lab"><font color="Green" title="E40">A5</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="E18:7"/><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="c15">P1</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="#E1:7"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E8:7"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="compts_1.html#T23" title="COMPTS_1:th.23">COMPTS_1:23</a></span>;<br/>
<a NAME="E19:7"/><span class="kw">then </span>
<font color="Maroon" title="c15">P1</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#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:7"/>
 <a href="topreal1.html#K1" title="TOPREAL1:func.1">R^2-unit_square</a>  is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E15:7"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="compts_1.html#T19" title="COMPTS_1:th.19">COMPTS_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
