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

<span class="kw">let </span><font color="Maroon" title="c1">eps</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> implies  for <font color="Olive" title="b1">d</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">d</font> holds <br/> ex <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp; <font color="Olive" title="b3">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r2</font> &amp; <font color="Olive" title="b3">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:32"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">d</font> holds <br/> ex <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp; <font color="Olive" title="b3">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r2</font> &amp; <font color="Olive" title="b3">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">eps1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:32"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">eps1</font>
 <span class="kw">and </span><br/><a NAME="E4:32"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">eps1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T5" onmouseover="rs('xreal_1/T5')" onmouseout="rh()">XREAL_1:5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">eps1</font> = <font color="Maroon" title="c2">eps1</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">d</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> implies  ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp; <font color="Olive" title="b2">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Olive" title="b2">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E6:32"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp; <font color="Olive" title="b2">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Olive" title="b2">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1"/>
( <font color="Maroon" title="c3">eps1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> or <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">eps1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1"/>
<font color="Maroon" title="c3">eps1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp; <font color="Olive" title="b2">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Olive" title="b2">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/><div class="add"><a NAME="E2:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a></span>;<br/><a NAME="E3:32_1_1"/>
<font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E4:32_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">c</font> being   <a href="xreal_0.html#NM1" title="XREAL_0: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="c4">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn2.html#T25" target="_self" onmouseover="rs('urysohn2/T25')" onmouseout="rh()">Th25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">r1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:32_1_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font>
 <span class="kw">and </span><br/><a NAME="E7:32_1_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">and </span><br/><a NAME="E8:32_1_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<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="E9:32_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#K10" title="PROB_1:func.10">halfline</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="urysohn1.html#D3" onmouseover="rs('urysohn1/D3')" onmouseout="rh()">URYSOHN1:def 3</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A8</font></span>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E11:32_1_1"/>
<font color="Maroon" title="c1">eps</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">eps</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E12:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E13:32_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">c</font> being   <a href="xreal_0.html#NM1" title="XREAL_0: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="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn2.html#T25" target="_self" onmouseover="rs('urysohn2/T25')" onmouseout="rh()">Th25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E15:32_1_1"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font>
 <span class="kw">and </span><br/><a NAME="E16:32_1_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:32_1_1"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<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/><a NAME="E18:32_1_1"/>
( <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#K10" title="PROB_1:func.10">halfline</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32_1_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="urysohn1.html#D3" onmouseover="rs('urysohn1/D3')" onmouseout="rh()">URYSOHN1:def 3</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E19:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A12</font></span>: 
( <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E20:32_1_1"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E15:32_1_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E21:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">r1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp; <font color="Olive" title="b1">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r2</font> &amp; <font color="Olive" title="b1">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c6">r2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K3" title="LIMFUNC1:func.3">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/><a NAME="E22:32_1_1"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">eps</font>
 ;<br/><a NAME="E23:32_1_1"/>
<font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">eps</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">eps1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:32_1_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E20:32_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E24:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E16:32_1_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E22:32_1_1"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="urysohn2.html#T30" target="_self" onmouseover="rs('urysohn2/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E25:32_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><span class="kw">hence </span><a NAME="E26:32_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#K3" title="LIMFUNC1:func.3">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E7:32_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E15:32_1_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E19:32_1_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E24:32_1_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xxreal_1.html#T233" onmouseover="rs('xxreal_1/T233')" onmouseout="rh()">XXREAL_1:233</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="32_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_2"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">eps1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp; <font color="Olive" title="b2">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Olive" title="b2">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">r1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_2"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E4:32_1_2"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_2"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="urysohn2.html#T25" target="_self" onmouseover="rs('urysohn2/T25')" onmouseout="rh()">Th25</a></span>;<br/><a NAME="E6:32_1_2"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
( <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#K10" title="PROB_1:func.10">halfline</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="urysohn1.html#D3" onmouseover="rs('urysohn1/D3')" onmouseout="rh()">URYSOHN1:def 3</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:32_1_2"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">eps1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_2"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E4:32_1_2"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E8:32_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">c</font> being   <a href="xreal_0.html#NM1" title="XREAL_0: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="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">eps1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn2.html#T25" target="_self" onmouseover="rs('urysohn2/T25')" onmouseout="rh()">Th25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E10:32_1_2"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font>
 <span class="kw">and </span><br/><a NAME="E11:32_1_2"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
<font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">eps1</font>
 <span class="kw">and </span><br/><a NAME="E12:32_1_2"/><span class="lab"><font color="Green" title="E32">A24</font></span>: 
<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/><span class="kw">take </span>
<font color="Maroon" title="c5">r1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp; <font color="Olive" title="b1">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r2</font> &amp; <font color="Olive" title="b1">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c6">r2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K3" title="LIMFUNC1:func.3">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )</span><br/><a NAME="E13:32_1_2"/><span class="lab"><font color="Green" title="E33">A25</font></span>: 
( <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#K10" title="PROB_1:func.10">halfline</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_1_2"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="urysohn1.html#D3" onmouseover="rs('urysohn1/D3')" onmouseout="rh()">URYSOHN1:def 3</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:32_1_2"/>
 not <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_2"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="xxreal_1.html#T233" onmouseover="rs('xxreal_1/T233')" onmouseout="rh()">XXREAL_1:233</a></span>;<br/><a NAME="E15:32_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A26</font></span>: 
( <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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_2"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E16:32_1_2"/>
 not <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">halfline</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E10:32_1_2"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="xxreal_1.html#T233" onmouseover="rs('xxreal_1/T233')" onmouseout="rh()">XXREAL_1:233</a></span>;<br/><a NAME="E17:32_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A27</font></span>: 
( <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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:32_1_2"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E18:32_1_2"/>
<font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">eps1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_1_2"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E19:32_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#K3" title="LIMFUNC1:func.3">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#K3" title="LIMFUNC1:func.3">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">eps</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:32_1_2"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E5:32_1_2"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E10:32_1_2"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E15:32_1_2"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E17:32_1_2"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
