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<b>let </b><font color="Maroon" title="c1">eps</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E24">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">eps</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">d</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E2:32"/><i><font color="Green" title="E25">A2</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> &amp; <font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">eps1</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="E4:32"/><i><font color="Green" title="E26">A3</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">eps1</font> &amp; <font color="Maroon" title="c3">eps1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">eps</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">eps1</font> = <font color="Maroon" title="c3">eps1</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1"/>
( <font color="Maroon" title="c4">eps1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> or <font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">eps1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>suppose </b></a><a NAME="E1:32_1_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c4">eps1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font>
 ;<br/><div class="add"><a NAME="E2:32_1_1"/>
<font color="Maroon" title="c2">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:32"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E3:32_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  &amp; <font color="Maroon" title="c2">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> )
 <b>by </b><i><a class="ref" href="urysohn2.html#T29" target="_self" title="URYSOHN2:th.29">Th29</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:32_1_1"/><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c2">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  )
 ;<br/><a NAME="E6:32_1_1"/>
<font color="Maroon" title="c1">eps</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">eps</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">eps</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:32"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E7:32_1_1"/><b>then </b>
<font color="Maroon" title="c2">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E8:32_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  &amp; <font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="urysohn2.html#T29" target="_self" title="URYSOHN2:th.29">Th29</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:32_1_1"/><i><font color="Green" title="E29">A6</font></i>: 
( <font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  )
 ;<br/><b>take </b>
<font color="Maroon" title="c5">r1</font>
;<br/><b>take </b>
<font color="Maroon" title="c6">r2</font>
;<br/><a NAME="E11:32_1_1"/><i><font color="Green" title="E30">A7</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:32_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E12:32_1_1"/>
( ( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12" title="PROB_1:func.12">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">r1</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 ) &amp; ( <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12" title="PROB_1:func.12">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="c6">r2</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 ) )
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:32_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="urysohn2.html#E10:32_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:32_1_1"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
( ( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">r1</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">r1</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 ) &amp; ( <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">r2</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="c6">r2</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 ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E14:32_1_1"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:32_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="urysohn2.html#E10:32_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:32_1_1"/><b>then </b><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn2.html#E10:32_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E16:32_1_1"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:32_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="urysohn2.html#E14:32_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:32_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">d</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">d</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">eps1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">eps</font>
 ;<br/><a NAME="E18:32_1_1"/><b>then </b><i><font color="Green" title="E35">A12</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">eps</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:32_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="urysohn2.html#E10:32_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="urysohn2.html#E15:32_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="urysohn2.html#E16:32_1_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="urysohn2.html#T34" target="_self" title="URYSOHN2:th.34">Th34</a></i>;<br/><a NAME="E19:32_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r1</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E14:32_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><b>hence </b><a NAME="E20:32_1_1"/>
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> &amp; <font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">eps</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:32_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="urysohn2.html#E10:32_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="urysohn2.html#E11:32_1_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="urysohn2.html#E13:32_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="urysohn2.html#E18:32_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a>, <a class="ref" href="urysohn1.html#T1" title="URYSOHN1:th.1">URYSOHN1:1</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2"><b>suppose </b></a><a NAME="E1:32_1_2"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">eps1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:32_1_2"/><i><font color="Green" title="E28">A14</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:32"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="urysohn2.html#T29" target="_self" title="URYSOHN2:th.29">Th29</a></i>;<br/><a NAME="E4:32_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">eps1</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:32_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="urysohn2.html#E3:32_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E5:32_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  &amp; <font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">eps1</font> )
 <b>by </b><i><a class="ref" href="urysohn2.html#T29" target="_self" title="URYSOHN2:th.29">Th29</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:32_1_2"/><i><font color="Green" title="E29">A15</font></i>: 
( <font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">eps1</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  )
 ;<br/><b>take </b>
<font color="Maroon" title="c5">r1</font>
;<br/><b>take </b>
<font color="Maroon" title="c6">r2</font>
;<br/><a NAME="E8:32_1_2"/><i><font color="Green" title="E30">A16</font></i>: 
not <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:32_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="urysohn1.html#T1" title="URYSOHN1:th.1">URYSOHN1:1</a></i>;<br/><a NAME="E9:32_1_2"/><i><font color="Green" title="E31">A17</font></i>: 
not <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:32"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="urysohn2.html#E7:32_1_2"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="urysohn1.html#T1" title="URYSOHN1:th.1">URYSOHN1:1</a></i>;<br/><a NAME="E10:32_1_2"/>
( ( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12" title="PROB_1:func.12">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">r1</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 ) &amp; ( <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12" title="PROB_1:func.12">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="c6">r2</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 ) )
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:32_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="urysohn2.html#E7:32_1_2"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:32_1_2"/><b>then </b><i><font color="Green" title="E32">A18</font></i>: 
( ( <font color="Maroon" title="c5">r1</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">r1</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 ) &amp; ( <font color="Maroon" title="c6">r2</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="c6">r2</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 ) )
 <b>by </b><i><a class="txt" href="urysohn2.html#E8:32_1_2"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="urysohn2.html#E9:32_1_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E12:32_1_2"/>
<font color="Maroon" title="c6">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">eps1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r1</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E7:32_1_2"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E13:32_1_2"/>
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font> &amp; <font color="Maroon" title="c2">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">eps</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:32"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="urysohn2.html#E3:32_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="urysohn2.html#E7:32_1_2"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="urysohn2.html#E11:32_1_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
