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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>;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E23">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>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1"/>
( ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> ) or ( 0 <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 ) or ( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>suppose </b></a><a NAME="E1:26_1_1"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_1"/>
( <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 or 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_1"><b>case </b></a><a NAME="E1:26_1_1_1_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">c</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="E3:26_1_1_1_1_1"/><i><font color="Green" title="E26">A4</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="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <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#E1:26"><i><font color="Green" title="E23">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">c</font> = <font color="Maroon" title="c3">c</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/><a NAME="E5:26_1_1_1_1_1"/>
<font color="Maroon" title="c4">c</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> 0
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:26_1_1_1_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="urysohn2.html#E3:26_1_1_1_1_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><a NAME="E6:26_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">c</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> 0</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> 
 <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="E7:26_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <b>by </b><i><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/><b>hence </b><a NAME="E8:26_1_1_1_1_1"/>
 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="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> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:26_1_1_1_1_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2"><b>case </b></a><a NAME="E1:26_1_1_1_1_2"/><i><font color="Green" title="E25">A5</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c3">a1</font> = 0;<br/><b>consider </b><font color="Maroon" title="c4">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:26_1_1_1_1_2"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c4">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; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</font> <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#E1:26_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="urysohn2.html#E1:26_1_1_1_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="urysohn2.html#T28" target="_self" title="URYSOHN2:th.28">Th28</a></i>;<br/><a NAME="E4:26_1_1_1_1_2"/>
<font color="Maroon" title="c4">c</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> 0</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> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:26_1_1_1_1_2"><i><font color="Green" title="E26">A6</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="E5:26_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <b>by </b><i><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="E6:26_1_1_1_1_2"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</font> <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#E1:26_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="urysohn2.html#E3:26_1_1_1_1_2"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E7:26_1_1_1_1_2"/>
 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="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> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:26_1_1_1_1_2"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:26_1_1"/>
 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="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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>suppose </b></a><a NAME="E1:26_1_2"/><i><font color="Green" title="E24">A8</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">b1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:26_1_2"/><i><font color="Green" title="E25">A9</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><b>consider </b><font color="Maroon" title="c5">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:26_1_2"/><i><font color="Green" title="E26">A10</font></i>: 
( <font color="Maroon" title="c5">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="c3">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">b1</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:26_1_2"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="urysohn2.html#T28" target="_self" title="URYSOHN2:th.28">Th28</a></i>;<br/><a NAME="E6:26_1_2"/><i><font color="Green" title="E27">A11</font></i>: 
<font color="Maroon" title="c5">c</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> 0</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> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:26_1_2"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">c</font>
;<br/><b>thus </b><a NAME="E7:26_1_2"/>
( <font color="Maroon" title="c5">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="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c5">c</font> <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#E1:26_1_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="urysohn2.html#E3:26_1_2"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="urysohn2.html#E5:26_1_2"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="urysohn2.html#E6:26_1_2"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3"><b>suppose </b></a><a NAME="E1:26_1_3"/>
( 0 <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 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:26_1_3"/><i><font color="Green" title="E24">A12</font></i>: 
( <font color="Maroon" title="c3">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="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <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#E1:26"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="urysohn2.html#T28" target="_self" title="URYSOHN2:th.28">Th28</a></i>;<br/><a NAME="E4:26_1_3"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c3">c</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> 0</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> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:26_1_3"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c3">c</font>
;<br/><b>thus </b><a NAME="E5:26_1_3"/>
( <font color="Maroon" title="c3">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="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <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#E3:26_1_3"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="urysohn2.html#E4:26_1_3"><i><font color="Green" title="E25">A13</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_4"><b>suppose </b></a><a NAME="E1:26_1_4"/><i><font color="Green" title="E24">A14</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_4_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_4_1_1_1"><b>case </b></a><a NAME="E1:26_1_4_1_1_1"/><i><font color="Green" title="E25">A15</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">c</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="E3:26_1_4_1_1_1"/><i><font color="Green" title="E26">A16</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="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <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#E1:26"><i><font color="Green" title="E23">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">c</font> = <font color="Maroon" title="c3">c</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/><a NAME="E5:26_1_4_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:26_1_4_1_1_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="urysohn2.html#E3:26_1_4_1_1_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:26_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">c</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
 ;<br/><a NAME="E7:26_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">c</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>
 <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="E8:26_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">c</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> 0</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span></span>)</span>
 <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="E9:26_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <b>by </b><i><a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/><b>hence </b><a NAME="E10:26_1_4_1_1_1"/>
 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="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> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:26_1_4_1_1_1"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_4_1_1_2"><b>case </b></a><a NAME="E1:26_1_4_1_1_2"/><i><font color="Green" title="E25">A17</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c3">b1</font> = 1;<br/><b>consider </b><font color="Maroon" title="c4">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:26_1_4_1_1_2"/><i><font color="Green" title="E26">A18</font></i>: 
( <font color="Maroon" title="c4">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="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="urysohn2.html#E1:26_1_4"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="urysohn2.html#E1:26_1_4_1_1_2"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="urysohn2.html#T28" target="_self" title="URYSOHN2:th.28">Th28</a></i>;<br/><a NAME="E4:26_1_4_1_1_2"/>
<font color="Maroon" title="c4">c</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> 0</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> 
 <b>by </b><i><a class="txt" href="urysohn2.html#E3:26_1_4_1_1_2"><i><font color="Green" title="E26">A18</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="E5:26_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <b>by </b><i><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="E6:26_1_4_1_1_2"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</font> <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#E1:26_1_4"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="urysohn2.html#E3:26_1_4_1_1_2"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:26_1_4_1_1_2"/>
 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="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> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E5:26_1_4_1_1_2"><i><font color="Green" title="E27">A19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:26_1_4"/>
 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="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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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