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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V10" title="PRE_TOPC:attr.10">normal</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> holds <br/>for <font color="Olive" title="b3">G</font> being   <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font><br/> for <font color="Olive" title="b4">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b5">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">r</font> &amp; <font color="Olive" title="b4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  holds <br/><font color="Olive" title="b5">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> implies for <font color="Olive" title="b1">G</font> being   <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  holds <br/><font color="Olive" title="b3">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  holds <br/><font color="Olive" title="b3">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">G</font> be    <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  holds <br/><font color="Olive" title="b2">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  holds <br/><font color="Olive" title="b1">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  implies <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:19"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">and </span><br/><a NAME="E3:19"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E4:19"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><div class="add"><a NAME="E2:19_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></span>;<br/><a NAME="E3:19_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:19_1_1"/>
<font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_2"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:19_1_2"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c4">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Maroon" title="c7">n</font> as    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c7">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></span>;<br/><a NAME="E5:19_1_2"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/>( <font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b1">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b2">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></span>;<br/><a NAME="E6:19_1_2"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E3:19_1_2"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_1_2_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_1_2"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="urysohn1.html#T5" title="URYSOHN1:th.5">URYSOHN1:5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c5">r</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></span>;<br/><a NAME="E3:19_1_2_1_1_1"/>
( 1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a>, <a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:19_1_2_1_1_1"/>
<font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E1:19_1_2_1_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:19_1_2_1_1_2"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><div class="add"><a NAME="E2:19_1_2_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1_2_1_1_2"/>
<font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:19_1_2"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E6:19_1_2"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E5:19_1_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E1:19_1_2_1_1_2"><span class="lab"><font color="Green" title="E25">A10</font></span></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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:19_1_2"/>
<font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3"><span class="kw">suppose </span></a><a NAME="E1:19_1_3"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><div class="add"><a NAME="E2:19_1_3"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></span>;<br/><a NAME="E3:19_1_3"/>
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E1:19_1_3"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:19_1_3"/>
<font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_3"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
