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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V5">being_T4</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>1</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:8"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <a href="urysohn1.html#K2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,0;<br/>
<a NAME="E5:8"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="urysohn1.html#K2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="urysohn3.html#T4" target="_self">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="urysohn3.html#E3:8"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="urysohn3.html#E3:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="urysohn3.html#E5:8"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="urysohn1.html#K2">DYADIC</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 ) &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>] means <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>];<br/>
<a NAME="E8:8"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:8_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:8"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="urysohn3.html#E2:8_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1<b> such that </b><br/><a NAME="E5:8_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="urysohn1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="urysohn3.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:8_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>11</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:8_1_1"/>
<font color="Maroon">c<sub>11</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:8_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:8_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_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="E1:8_1_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E1:8_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>14</sub></font> = 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/><a NAME="E2:8_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/><a NAME="E3:8_1_1_1_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="urysohn1.html#D3">URYSOHN1:def 3</a></i>;<br/><a NAME="E4:8_1_1_1_1_1"/>
not 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1_1_1_1_1"/>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:8_1_1_1_1_1_1"/><i><font color="Green">E18</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="urysohn1.html#D3">URYSOHN1:def 3</a></i>;<br/>
<a NAME="E4:8_1_1_1_1_1_1"/><i><font color="Green">E19</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E5:8_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E3:8_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xcmplx_1.html#T96">XCMPLX_1:96</a></i>;<br/>
<a NAME="E6:8_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E4:8_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E7:8_1_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="urysohn3.html#E5:8_1_1_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E8:8_1_1_1_1_1_1"/>
2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:8_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pepin.html#T71">PEPIN:71</a></i>;<br/>
<a NAME="E9:8_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="urysohn3.html#E4:8_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E6:8_1_1_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/>
<a NAME="E10:8_1_1_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:8_1_1_1_1_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:8_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 
<b>by </b><i><a class="txt" href="urysohn3.html#E9:8_1_1_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn3.html#E1:8_1_1_1_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1_1_1_1_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:8_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:8_1_1_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:8_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:8_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="urysohn3.html#E3:8_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>14</sub></font> = 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>;<br/><a NAME="E2:8_1_1_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/><a NAME="E3:8_1_1_1_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="urysohn1.html#D3">URYSOHN1:def 3</a></i>;<br/><a NAME="E4:8_1_1_1_1_2"/>
not 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1_1_1_2_1"/>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:8_1_1_1_1_2_1"/><i><font color="Green">E18</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font> &amp; 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="urysohn1.html#D3">URYSOHN1:def 3</a></i>;<br/>
<a NAME="E4:8_1_1_1_1_2_1"/><i><font color="Green">E19</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E5:8_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E20</font></i>: 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E3:8_1_1_1_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xcmplx_1.html#T96">XCMPLX_1:96</a></i>;<br/>
<a NAME="E6:8_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E4:8_1_1_1_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E7:8_1_1_1_1_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="urysohn3.html#E5:8_1_1_1_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<a NAME="E8:8_1_1_1_1_2_1"/>
2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:8_1_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pepin.html#T71">PEPIN:71</a></i>;<br/>
<a NAME="E9:8_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="urysohn3.html#E4:8_1_1_1_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E6:8_1_1_1_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/>
<a NAME="E10:8_1_1_1_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:8_1_1_1_1_2_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:8_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 
<b>by </b><i><a class="txt" href="urysohn3.html#E9:8_1_1_1_1_2_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn3.html#E1:8_1_1_1_1_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1_1_1_2_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:8_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:8_1_1_1_1_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:8_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:8_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="urysohn3.html#E3:8_1_1_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:8_1_1"/>
<font color="Maroon">c<sub>12</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:8_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> holds <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font>;<br/>

<a NAME="E1:8_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:8_1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<a NAME="E3:8_1_2"/>
 <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="urysohn1.html#T11">URYSOHN1:11</a></i>;<br/>
<b>hence </b><a NAME="E4:8_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E5:8_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="urysohn3.html#E2:8_1_2"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="urysohn1.html#K2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="urysohn3.html#T4" target="_self">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<a NAME="E10:8_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <a href="urysohn1.html#K2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="urysohn3.html#T4" target="_self">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="urysohn3.html#E3:8"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E12:8_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_3"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="urysohn1.html#K2">DYADIC</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="partfun1.html#T121">PARTFUN1:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="urysohn1.html#K2">DYADIC</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="urysohn3.html#E10:8_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>thus </b><a NAME="E3:8_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>16</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>16</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>17</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> holds <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E6:8_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="urysohn3.html#E7:8_1"><i><font color="Green">E14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E14:8_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font>]
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E15:8_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E14:8_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:8"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 
<b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="urysohn3.html#E8:8"><i><font color="Green">E10</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E11:8"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="urysohn1.html#K2">DYADIC</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="urysohn1.html#K2">DYADIC</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>] means ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>2</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>2</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>3</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>2</sub></font> holds <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) );<br/>
<a NAME="E12:8"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is   <a href="funct_1.html#NM1">Function</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E13:8"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <a href="urysohn1.html#K2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E3:8"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E14:8"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <a href="urysohn1.html#K2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:8"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> is   <a href="funct_1.html#NM1">Function</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <a href="urysohn1.html#K2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E12:8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="seqfunc.html#M1">Functional_Sequence</a> of  <a href="urysohn1.html#K2">DYADIC</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="seqfunc.html#D1">SEQFUNC:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) );<br/>
<a NAME="E17:8"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="partfun1.html#NM1">PartFunc</a> of  <a href="urysohn1.html#K2">DYADIC</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>4</sub></font>[0,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="urysohn3.html#E11:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E18:8"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[0]
 
<b>by </b><i><a class="txt" href="urysohn3.html#E11:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E19:8"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>10</sub></font>]
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="urysohn1.html#K2">DYADIC</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:8_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E11:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:8_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:8_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="urysohn3.html#E3:8_2"><i><font color="Green">E16</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>

<b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="urysohn1.html#K2">DYADIC</a> , <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:8_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E11:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:8_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E6:8_2"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:8"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="urysohn3.html#E18:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="urysohn3.html#E19:8"><i><font color="Green">E14</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E21:8"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E20:8"><i><font color="Green">E15</font></i></a></i>;<br/>


</div>
