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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c1">n</font> holds <br/>(  <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="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c1">n</font> implies (  <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">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:9"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:9"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
(  <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">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">and </span><br/><a NAME="E4:9"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D3" target="_self" title="URYSOHN1:def.3">Def3</a></span>;<br/>
<a NAME="E5:9"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c3">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:9"/>
(  <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">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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