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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">ADD</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:48"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font> holds  <font color="Maroon">ADD</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f1</font>,<font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Olive">f2</font>
 ;<br/>

<b>set </b><font color="Maroon">F</font> =  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<a href="numbers.html#K1">REAL</a> ;<br/>
<b>let </b><font color="Maroon">map'</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E2:48"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">map'</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">ADD</font>
 ;<br/>

<a NAME="E3:48"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Maroon">map'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/>
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st <font color="Maroon">map'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">map'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>
<a NAME="E4:48_1"/>
<font color="Maroon">ADD</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">map'</font>,<font color="Maroon">map'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">map'</font>
 
<b>by </b><i>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<a NAME="E5:48_1"/><b>then </b>
<font color="Maroon">map'</font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">map'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">map'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:48_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">map'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">map'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">map'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:48_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">map''</font> = <font color="Maroon">map'</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:48"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> holds  <font color="Maroon">map''</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">map''</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">mCla</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>



<b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">mCla</font> <a href="hidden.html#R2">in</a> <font color="Maroon">map''</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">Cla</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">Cla</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">map'</font> &amp; <font color="Maroon">Cla</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp; <font color="Maroon">mCla</font> <a href="hidden.html#R1">=</a> <font color="Maroon">map''</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Cla</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Cla</font> = <font color="Maroon">Cla</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E5:48_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:48_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E9:48_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">map'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:48_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">map'</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">map'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:48_2"/><b>then </b>
( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">map'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> &amp; 0 <a href="hidden.html#R1">=</a> <font color="Maroon">mCla</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E12:48_2"/><b>then </b>
( <font color="Maroon">mCla</font> <a href="hidden.html#R2">in</a> <font color="Maroon">map'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">map''</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">map''</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E13:48_2"/>
<font color="Maroon">mCla</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">map''</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:48"/><b>then </b>
<font color="Maroon">map''</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T57">TOPS_2:57</a></i>;<br/>
<b>hence </b><a NAME="E7:48"/>
<font color="Maroon">map'</font> is <a href="pscomp_1.html#V9">continuous</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>


</div>
