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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>assume </b><a NAME="E1:112_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">Y</font> is <a href="rcomp_1.html#V3">open</a>
 ;<br/><b>set </b><font color="Maroon">p1Y</font> = <a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:112_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font>
 ;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon">p1</font> = <a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>;<br/><a NAME="E3:112_1_1_1_1_1"/>
<a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:112_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E6:112_1_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="rcomp_1.html#T40">RCOMP_1:40</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E8:112_1_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E9:112_1_1_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E10:112_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#T6" target="_self">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon">Q</font> = <a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <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/><b>take </b><font color="Maroon">Q</font> = <font color="Maroon">Q</font>;<br/><a NAME="E12:112_1_1_1_1_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">q3</font>,<font color="Olive">p3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">q3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">p3</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q3</font> &amp; <font color="Olive">q3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:112_1_1_1_1_1"/>
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E14:112_1_1_1_1_1"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T4" target="_self">Th4</a>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/><b>thus </b><a NAME="E15:112_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:112_1_1_1_1"/>
 ex <font color="Olive">Q</font> being  <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> st <br/>( <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q</font> <a href="tarski.html#R1">c=</a> <a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> )
 ;<br/><b>hence </b><a NAME="E3:112_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:112_1_1_1"/>
<a href="pscomp_1.html#K13" target="_self">proj1</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57">TOPS_1:57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:112_1_1"/>
 <a href="pscomp_1.html#K13" target="_self">proj1</a>  is <a href="pscomp_1.html#V9" target="_self">continuous</a>
 <b>by </b><i/>;<br/>


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