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

<b>let </b><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25_1_1"/><i><font color="Green" title="E22">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( (  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  implies <font color="Maroon" title="c5">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( for <font color="Olive" title="b2">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">R</font> holds <br/><font color="Maroon" title="c5">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b2">S</font> ) )
 <b>and </b><br/><a NAME="E2:25_1_1"/><i><font color="Green" title="E23">A2</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( (  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  implies <font color="Maroon" title="c6">G2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( for <font color="Olive" title="b2">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">R</font> holds <br/><font color="Maroon" title="c6">G2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b2">S</font> ) )
 ;<br/>

<a NAME="E3:25_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Maroon" title="c5">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_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="E1:25_1_1_1_1"/>
(  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  or  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c5">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> = 
0
<b>by </b><i><a class="txt" href="urysohn3.html#E1:25_1_1"><i><font color="Green" title="E22">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">G2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="urysohn3.html#E2:25_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="urysohn3.html#E1:25_1_1_1_1_1"><i><font color="Green" title="E24">A3</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:25_1_1_1_1_1"/>
<font color="Maroon" title="c5">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_1_1_2"/>
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">S</font> =  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c4">R</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  ;<br/><font color="Maroon" title="c5">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> = 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c9">S</font>
<b>by </b><i><a class="txt" href="urysohn3.html#E1:25_1_1"><i><font color="Green" title="E22">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">G2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="urysohn3.html#E2:25_1_1"><i><font color="Green" title="E23">A2</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:25_1_1_1_1_2"/>
<font color="Maroon" title="c5">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:25_1_1"/>
<font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>


</div>
