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

<b>thus </b><a NAME="E1:30_1_1"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="rltopsp1.html#V6">add-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">x2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D8">RLTOPSP1:def 8</a><br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:30_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">z12</font> = <font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" target="_self">MetricSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" target="_self">MetricSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" target="_self">MetricSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:30_1_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, </i>;<br/>
<a NAME="E6:30_1_1_1"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:30_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z12</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z1</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>, <font color="Maroon">V2</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z2</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E10:30_1_1_1"/>
(  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z1</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z2</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/>
<a NAME="E11:30_1_1_1"/><b>then </b>
(  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z1</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z2</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E12:30_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">V1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V2</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">r2</font> = <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E13:30_1_1_1"/><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E14:30_1_1_1"/>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">z1</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">z2</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>
<a NAME="E15:30_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> )
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E16:30_1_1_1"/>
<font color="Maroon">V1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</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="E1:30_1_1_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> <a href="rusub_4.html#K6">+</a> <font color="Maroon">V2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E3:30_1_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">u1</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" target="_self">MetricSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" target="_self">MetricSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">zz12</font> = <font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x2</font>, <font color="Maroon">zz1</font> = <font color="Maroon">x1</font>, <font color="Maroon">zz2</font> = <font color="Maroon">x2</font>, <font color="Maroon">vv1</font> = <font color="Maroon">v</font>, <font color="Maroon">uu1</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ww1</font> = <font color="Maroon">w</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<i><font color="Green">E47</font></i>:  <a href="metric_1.html#K4">dist</a> <font color="Maroon">z12</font>,<font color="Maroon">w1</font> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz12</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">ww1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">zz1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">zz2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">ww1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">zz1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">zz2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">vv1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">uu1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">zz1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">zz2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">uu1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">vv1</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T45">RLVECT_1:45</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">zz1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">zz2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">vv1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">uu1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">zz1</font> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">vv1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">zz2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">uu1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">zz1</font> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">vv1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">zz2</font> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">uu1</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E9:30_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">zz1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">zz2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">uu1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">uu1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/>
<a NAME="E10:30_1_1_1_1"/><i><font color="Green">E51</font></i>: 
( <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">z1</font>,<font color="Maroon">v1</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">uu1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">z2</font>,<font color="Maroon">u1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E11:30_1_1_1_1"/>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">z1</font>,<font color="Maroon">v1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">z2</font>,<font color="Maroon">u1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E12:30_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">z1</font>,<font color="Maroon">v1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">z2</font>,<font color="Maroon">u1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E13:30_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">z12</font>,<font color="Maroon">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E14:30_1_1_1_1"/><b>then </b>
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z12</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E15:30_1_1_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:30_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="rusub_4.html#K6">+</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E2:30_1_1"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="rltopsp1.html#V7">Mult-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D9">RLTOPSP1:def 9</a><br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:30_1_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font>, <font color="Maroon">az</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" target="_self">MetricSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E4:30_1_1_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, </i>;<br/>
<a NAME="E5:30_1_1_2"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">r0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:30_1_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">az</font>,<font color="Maroon">r0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> = <font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2;<br/>
<b>set </b><font color="Maroon">r2</font> =  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1;<br/>
<b>reconsider </b><font color="Maroon">W</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:30_1_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/>
<a NAME="E10:30_1_1_2"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:30_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E12:30_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E13:30_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T35">XREAL_1:35</a></i>;<br/>
<a NAME="E14:30_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T41">XREAL_1:41</a></i>;<br/>
<a NAME="E15:30_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T78">XREAL_1:78</a>, </i>;<br/>
<a NAME="E16:30_1_1_2"/><b>then </b>
( <font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font> )
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E17:30_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E18:30_1_1_2"/><b>then </b>
0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a>, </i>;<br/>
<a NAME="E19:30_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1
 
<b>by </b><i><a class="ref" href="square_1.html#T38">SQUARE_1:38</a></i>;<br/>
<a NAME="E20:30_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>
<a NAME="E21:30_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E22:30_1_1_2"/>
for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1 holds <br/><font color="Olive">s</font> <a href="convex1.html#K1">*</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:30_1_1_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1
 ;<br/>

<a NAME="E2:30_1_1_2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1
 
<b>by </b><i><a class="ref" href="complex1.html#T146">COMPLEX1:146</a></i>;<br/>
<b>thus </b><a NAME="E3:30_1_1_2_1"/>
<font color="Maroon">s</font> <a href="convex1.html#K1">*</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</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="E1:30_1_1_2_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="convex1.html#K1">*</a> <font color="Maroon">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E3:30_1_1_2_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" target="_self">MetricSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" target="_self">MetricSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">zza</font> = <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font>, <font color="Maroon">zz</font> = <font color="Maroon">x</font>, <font color="Maroon">vv1</font> = <font color="Maroon">v</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ww1</font> = <font color="Maroon">w</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<i><font color="Green">E67</font></i>:  <a href="metric_1.html#K4">dist</a> <font color="Maroon">az</font>,<font color="Maroon">w1</font> = 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zza</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">ww1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">ww1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, </i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T48">RLVECT_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T49">RLVECT_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/>
<a NAME="E10:30_1_1_2_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T7">NORMSP_1:7</a></i>;<br/>
<a NAME="E11:30_1_1_2_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/>
<a NAME="E12:30_1_1_2_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/>
<a NAME="E13:30_1_1_2_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">zz</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">vv1</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T146">COMPLEX1:146</a></i>;<br/>
<a NAME="E14:30_1_1_2_1_1"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">v1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:30_1_1_2_1_1"/><i><font color="Green">E72</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">v1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E16:30_1_1_2_1_1"/><i><font color="Green">E77</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E17:30_1_1_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E18:30_1_1_2_1_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/>
<a NAME="E19:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E20:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, </i>;<br/>
<a NAME="E21:30_1_1_2_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T76">XCMPLX_1:76</a></i>;<br/>
<a NAME="E22:30_1_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E23:30_1_1_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T41">XREAL_1:41</a></i>;<br/>
<a NAME="E24:30_1_1_2_1_1"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E25:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T185">XREAL_1:185</a></i>;<br/>
<a NAME="E26:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, </i>;<br/>
<a NAME="E27:30_1_1_2_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<font color="Maroon">zz</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">vv1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">zz</font></span>)</span> = 
<font color="Maroon">vv1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<font color="Maroon">vv1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">vv1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/>
<a NAME="E29:30_1_1_2_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">zz</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vv1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">zz</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/>
<a NAME="E30:30_1_1_2_1_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1 <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/>
<a NAME="E31:30_1_1_2_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E32:30_1_1_2_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vv1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">zz</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="normsp_1.html#T11">NORMSP_1:11</a></i>;<br/>
<a NAME="E33:30_1_1_2_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">zz</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vv1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">zz</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">zz</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E34:30_1_1_2_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">zz</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E35:30_1_1_2_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a>, <a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E36:30_1_1_2_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">zz</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/>
<a NAME="E37:30_1_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/>
<a NAME="E38:30_1_1_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T41">XREAL_1:41</a></i>;<br/>
<a NAME="E39:30_1_1_2_1_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/>
<a NAME="E40:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p5"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">zz</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">zz</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E41:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E42:30_1_1_2_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T76">XCMPLX_1:76</a></i>;<br/>
<a NAME="E43:30_1_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E44:30_1_1_2_1_1"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E45:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T185">XREAL_1:185</a></i>;<br/>
<a NAME="E46:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p4"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">z1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, </i>;<br/>
<a NAME="E47:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E48:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">zz</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">vv1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">vv1</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E49:30_1_1_2_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">az</font>,<font color="Maroon">w1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r0</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E50:30_1_1_2_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">az</font>,<font color="Maroon">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E51:30_1_1_2_1_1"/><b>then </b>
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">az</font>,<font color="Maroon">r0</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E52:30_1_1_2_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:30_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> or <font color="Olive">b<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) ) )
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E3:30_1_1"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="pre_topc.html#V2">TopSpace-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">LTSN</font> =  <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font>;<br/>
<a NAME="E1:30_1_1_3"/><i><font color="Green">E91</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:30_1_1_3"/><i><font color="Green">E92</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:30_1_1_3"/><b>then </b><i><font color="Green">E93</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E4:30_1_1_3"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E5:30_1_1_3"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> holds <br/><font color="Olive">a</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:30_1_1_3"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="pre_topc.html#V2">TopSpace-like</a>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E4:30_1_1"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">w1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v1</font>

<br/>.= 
<font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E5:30_1_1"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V4">add-associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a><br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font>, <font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E2:30_1_1_5"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font> &amp; <font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x1</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>


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

<b>thus </b><a NAME="E6:30_1_1"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V5">right_zeroed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E2:30_1_1_6"/>
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span></span>)</span> = 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E7:30_1_1"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V6">right_complementable</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">w</font> =  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v1</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E8:30_1_1"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V7">RealLinearSpace-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_8"><b>proof </b></a><div class="add">

<div><i><font color="Green">E96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E2:30_1_1_8_1"/>
( <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font> )
 <b>by </b><i/>;<br/><a NAME="E3:30_1_1_8_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:30_1_1_8_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E97</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E2:30_1_1_8_2"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font> )
 <b>by </b><i/>;<br/><a NAME="E3:30_1_1_8_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:30_1_1_8_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_8_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E2:30_1_1_8_3"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font>
 <b>by </b><i/>;<br/><a NAME="E3:30_1_1_8_3"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:30_1_1_8_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><b>hence </b><a NAME="E5:30_1_1_8_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_8_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>thus </b>1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:30_1_1_8"/>
 <a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font> is <a href="rlvect_1.html#V7">RealLinearSpace-like</a>
 <b>by </b><i>, , , <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>


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


</div>
