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

<b>set </b><font color="Maroon" title="c1">ZS</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">O</font> = 0 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = <font color="Maroon" title="c2">O</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">F</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span><b> such that </b><br/><a NAME="E3:60_1_1"/><i><font color="Green" title="E37">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = <font color="Maroon" title="c2">O</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span><b> such that </b><br/><a NAME="E5:60_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <font color="Maroon" title="c4">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon" title="c5">T</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<a NAME="E6:60_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>

<a NAME="E2:60_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:60_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:60_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">T</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<b>take </b><font color="Maroon" title="c7">W</font> =  <a href="rltopsp1.html#G1" title="RLTOPSP1:aggr.1">RLTopStruct</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c6">T</font> #);<br/>

<b>thus </b><a NAME="E8:60_1_1"/>
<font color="Maroon" title="c7">W</font> is <a href="rltopsp1.html#V5" title="RLTOPSP1:attr.5">strict</a>
 ;<br/>

<b>thus </b><a NAME="E9:60_1_1"/>
<font color="Maroon" title="c7">W</font> is <a href="rltopsp1.html#V6" title="RLTOPSP1:attr.6">add-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D8" target="_self" title="RLTOPSP1:def.8">RLTOPSP1:def 8</a><br/><b>let </b><font color="Maroon" title="c10">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">W</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:60_1_1_2"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c10">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:60_1_1_2"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">V</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">V1</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">O</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c12">V2</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">O</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">W</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c11">V1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c12">V2</font>
;<br/>

<a NAME="E4:60_1_1_2"/>
( <font color="Maroon" title="c11">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font> &amp; <font color="Maroon" title="c12">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:60_1_1_2"/>
( <font color="Maroon" title="c11">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>

<b>thus </b><a NAME="E6:60_1_1_2"/>
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V1</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">V2</font> )
 ;<br/>

<a NAME="E7:60_1_1_2"/>
<font color="Maroon" title="c10">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:60_1_1_2"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E8:60_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:60_1_1_2"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:60_1_1_2"/>
<font color="Maroon" title="c11">V1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c12">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">V</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E10:60_1_1"/>
<font color="Maroon" title="c7">W</font> is <a href="rltopsp1.html#V7" title="RLTOPSP1:attr.7">Mult-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D9" target="_self" title="RLTOPSP1:def.9">RLTOPSP1:def 9</a><br/><b>let </b><font color="Maroon" title="c9">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>;<br/><b>let </b><font color="Maroon" title="c10">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">W</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:60_1_1_3"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c10">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:60_1_1_3"/><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">V</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">r</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">V'</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">O</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">W</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c11">r</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c12">V'</font>
;<br/>

<a NAME="E5:60_1_1_3"/>
<font color="Maroon" title="c12">V'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:60_1_1_3"/>
<font color="Maroon" title="c12">V'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>

<b>thus </b><a NAME="E7:60_1_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">V'</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c13">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E8:60_1_1_3"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">r</font>
 ;<br/>

<a NAME="E9:60_1_1_3"/>
<font color="Maroon" title="c10">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">T</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:60_1_1_3"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:60_1_1_3"/><b>then </b>
( <font color="Maroon" title="c10">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c10">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E11:60_1_1_3"/>
<font color="Maroon" title="c13">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c12">V'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:60_1_1_3"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E11:60_1_1"/>
<font color="Maroon" title="c7">W</font> is <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:60_1_1_4"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c8">a</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c7">W</font>;<br/><b>assume </b><a NAME="E1:60_1_1_4_1"/>
<font color="Maroon" title="c8">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">W</font>
 ;<br/><a NAME="E2:60_1_1_4_1"/><b>then </b>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T42" title="ZFMISC_1:th.42">ZFMISC_1:42</a></i>;<br/><a NAME="E3:60_1_1_4_1"/><b>then </b>
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a>, <a class="ref" href="zfmisc_1.html#T93" title="ZFMISC_1:th.93">ZFMISC_1:93</a></i>;<br/><b>hence </b><a NAME="E4:60_1_1_4_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">W</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:60_1_1_4"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">W</font>
 <b>and </b><br/><a NAME="E4:60_1_1_4"/><i><font color="Green" title="E40">A8</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">W</font>
 ;<br/>

<a NAME="E5:60_1_1_4"/>
( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; ( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:60_1_1_4"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="rltopsp1.html#E4:60_1_1_4"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:60_1_1_4"/>
<font color="Maroon" title="c8">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E12:60_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font> holds  <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b1">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">Y</font> = <font color="Maroon" title="c9">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<a NAME="E2:60_1_1_5"/>
( <font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">X</font>,<font color="Maroon" title="c11">Y</font>) &amp; <font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">Y</font>,<font color="Maroon" title="c10">X</font>) )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:60_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_1_5"/>
<font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c8">x</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E13:60_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">z</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">z</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c11">X</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c12">Y</font> = <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c13">Z</font> = <font color="Maroon" title="c10">z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<a NAME="E2:60_1_1_6"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c12">Y</font>),<font color="Maroon" title="c13">Z</font>) &amp; <font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">Y</font>,<font color="Maroon" title="c13">Z</font>)) )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:60_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c10">z</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E14:60_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font> holds  <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<a NAME="E2:60_1_1_7"/>
<font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">X</font>,<font color="Maroon" title="c2">O</font>)
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:60_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_1_7"/>
<font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E15:60_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>  ex <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font> st <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c7">W</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c2">O</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c9">y</font>
;<br/>

<b>thus </b><a NAME="E2:60_1_1_8"/>
<font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c7">W</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:60_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E16:60_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b3">y</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c11">X</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c12">Y</font> = <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">y</font></span>)</span> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">a</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c12">Y</font>))
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:60_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X</font>,<font color="Maroon" title="c12">Y</font>)
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:60_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span>)</span>
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:60_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E3:60_1_1_9"/>
<font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">y</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E17:60_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b3">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c10">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>;<br/>



<b>set </b><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c8">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c9">b</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">X</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<a NAME="E2:60_1_1_10"/><i><font color="Green" title="E39">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c12">X</font>)
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:60_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<a NAME="E3:60_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">X</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c12">X</font>))
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:60_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:60_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:60_1_1_10"><i><font color="Green" title="E39">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E18:60_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b3">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c10">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>;<br/>



<b>set </b><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c8">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">b</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">X</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<a NAME="E2:60_1_1_11"/><i><font color="Green" title="E39">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c12">X</font>)
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:60_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<a NAME="E3:60_1_1_11"/>
<font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">a</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c12">X</font>))
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:60_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:60_1_1_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:60_1_1_11"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E19:60_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font> holds  1 <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c7">W</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">A'</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E3:60_1_1_12"/>
1 <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c9">X</font>)
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:60_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:60_1_1_12"/>
1 <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>


</div>
