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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> &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="c1">a</font> &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 implies  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#K2" title="URYSOHN1:func.2">DYADIC</a>  &amp; <font color="Maroon" title="c1">a</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">b</font> ) )</span><br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> &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="c1">a</font> &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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#K2" title="URYSOHN1:func.2">DYADIC</a>  &amp; <font color="Maroon" title="c1">a</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">b</font> )</span><br/>

<b>set </b><font color="Maroon" title="c3">eps</font> = <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>;<br/>
<a NAME="E2:25"/>
 <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">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E4:25"/><i><font color="Green" title="E23">A2</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="urysohn2.html#T23" target="_self" title="URYSOHN2:th.23">Th23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">aa</font> = <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">a</font>;<br/>
<b>set </b><font color="Maroon" title="c6">bb</font> = <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font>;<br/>
<a NAME="E5:25"/><i><font color="Green" title="E24">A3</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E4:25"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/>
<a NAME="E6:25"/><i><font color="Green" title="E25">A4</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> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<a NAME="E7:25"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="real_2.html#T121" title="REAL_2:th.121">REAL_2:121</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">m</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> <b> such that </b><br/><a NAME="E9:25"/><i><font color="Green" title="E26">A5</font></i>: 
( <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:25"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="urysohn2.html#T24" target="_self" title="URYSOHN2:th.24">Th24</a></i>;<br/>
<a NAME="E10:25"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="urysohn2.html#E6:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="urysohn2.html#E9:25"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="real_2.html#T178" title="REAL_2:th.178">REAL_2:178</a></i>;<br/>
<a NAME="E11:25"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="urysohn2.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:25"/><b>then </b><i><font color="Green" title="E27">A6</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="c7">m</font> &amp; <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font> )
 
<b>by </b><i><a class="txt" href="urysohn2.html#E6:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="real_2.html#T117" title="REAL_2:th.117">REAL_2:117</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>;<br/>
<a NAME="E13:25"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <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="c4">n</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E12:25"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="urysohn1.html#D3" title="URYSOHN1:def.3">URYSOHN1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> )</span><br/>

<b>thus </b><a NAME="E14:25"/>
( <font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c7">m</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E6:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="urysohn2.html#E9:25"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="urysohn2.html#E13:25"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="real_2.html#T178" title="REAL_2:th.178">REAL_2:178</a>, <a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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