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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>assume </b><a NAME="E1:110_2_1_1"/><i><font color="Green" title="E70">A7</font></i>: 
<font color="Maroon" title="c1">Y</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 ;<br/><b>set </b><font color="Maroon" title="c2">p1Y</font> = <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c1">Y</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:110_2_1_1_1_1"/><i><font color="Green" title="E71">A8</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c1">Y</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon" title="c5">p1</font> = <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font>;<br/><a NAME="E3:110_2_1_1_1_1"/>
<a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:110_2_1_1_1_1"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">g</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:110_2_1_1_1_1"/><i><font color="Green" title="E72">A9</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">g</font>
 <b>and </b><br/><a NAME="E6:110_2_1_1_1_1"/><i><font color="Green" title="E73">A10</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:110_2_1_1"><i><font color="Green" title="E70">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T40" title="RCOMP_1:th.40">RCOMP_1:40</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c6">g</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E8:110_2_1_1_1_1"/><i><font color="Green" title="E74">A11</font></i>: 
<a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:110_2_1_1_1_1"><i><font color="Green" title="E72">A9</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E9:110_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E10:110_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E75">A12</font></i>: 
<a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:110_2_1_1_1_1"><i><font color="Green" title="E74">A11</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">Q</font> = <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">g</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as   <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/><b>take </b><font color="Maroon" title="c9">Q</font> = <font color="Maroon" title="c8">Q</font>;<br/><a NAME="E12:110_2_1_1_1_1"/>
<font color="Maroon" title="c9">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">q3</font>,<font color="Olive" title="b2">p3</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">q3</font>, <font color="Olive" title="b2">p3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p3</font> &amp; <font color="Olive" title="b2">p3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T67" target="_self" title="PSCOMP_1:th.67">Th67</a></i>;<br/><b>hence </b><a NAME="E13:110_2_1_1_1_1"/>
<font color="Maroon" title="c9">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T68" target="_self" title="PSCOMP_1:th.68">Th68</a></i>;<br/><b>thus </b><a NAME="E14:110_2_1_1_1_1"/>
<font color="Maroon" title="c9">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E6:110_2_1_1_1_1"><i><font color="Green" title="E73">A10</font></i></a>, <a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a></i>;<br/><b>thus </b><a NAME="E15:110_2_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E10:110_2_1_1_1_1"><i><font color="Green" title="E75">A12</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:110_2_1_1_1"/>
 ex <font color="Olive" title="b1">Q</font> being  <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> st <br/>( <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> )
 ;<br/><b>hence </b><a NAME="E3:110_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c1">Y</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:110_2_1_1"/>
<a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c1">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57" title="TOPS_1:th.57">TOPS_1:57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:110_2_1"/>
 <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T54" target="_self" title="PSCOMP_1:th.54">Th54</a></i>;<br/>


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