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

<b>thus </b><a NAME="E1:12_1_1"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="normsp_1.html#V2">RealNormSpace-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span><br/> for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( ( <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> implies <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0 ) &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Olive">a</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</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="Olive">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">y</font></span><a href="normsp_1.html#K1">.||</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>proof </b></a><div class="add">

<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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>thus </b><a NAME="E1:12_1_1_1_1"/>
( <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E2:12_1_1_1_1_1"/>
( <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">x1</font></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T10">EUCLID:10</a>, <a class="ref" href="euclid.html#T11">EUCLID:11</a></i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_1_1_1"/>
( <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E2:12_1_1_1_1"/>
<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> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E2:12_1_1_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><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> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">x1</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14">EUCLID:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E3:12_1_1_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</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">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">y</font></span><a href="normsp_1.html#K1">.||</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E2:12_1_1_1_1_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">y1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:12_1_1_1_1_3"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">y1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:12_1_1_1_1_3"/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">y1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">x1</font></span><a href="euclid.html#K12">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y1</font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T15">EUCLID:15</a></i>;<br/>
<a NAME="E5:12_1_1_1_1_3"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">y1</font></span>)</span></span><a href="euclid.html#K12">.|</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">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y1</font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:12_1_1_1_1_3"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</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">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">y</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, </i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1_1_1"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="normsp_1.html#V2">RealNormSpace-like</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:12_1_1"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V7">RealLinearSpace-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>proof </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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>thus </b><a NAME="E1:12_1_1_2_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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1_1"><b>proof </b></a><div class="add">

<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="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E2:12_1_1_2_1_1"/><i><font color="Green">E66</font></i>: 
<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="euclid.html#K9">*</a> <font color="Maroon">v1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:12_1_1_2_1_1"/><i><font color="Green">E67</font></i>: 
<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="euclid.html#K9">*</a> <font color="Maroon">w1</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font>, <font color="Maroon">w2</font> = <font color="Maroon">w1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<b>thus </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> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<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>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T73">RVSUM_1:73</a></i>
<br/>.= 
<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>, , </i>
;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:12_1_1_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="binop_2.html#K9">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_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>, <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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>thus </b><a NAME="E1:12_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K9">+</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E2:12_1_1_2_2_1"/><i><font color="Green">E71</font></i>: 
<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="euclid.html#K9">*</a> <font color="Maroon">v1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:12_1_1_2_2_1"/><i><font color="Green">E72</font></i>: 
<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="euclid.html#K9">*</a> <font color="Maroon">v1</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">v</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T72">RVSUM_1:72</a></i>
<br/>.= 
<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>, , </i>
;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:12_1_1_2"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="binop_2.html#K11">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_3"><b>proof </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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>thus </b><a NAME="E1:12_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E2:12_1_1_2_3_1"/><i><font color="Green">E74</font></i>: 
<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="euclid.html#K9">*</a> <font color="Maroon">v1</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">v</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">b</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>
;<br/>
<b>hence </b><a NAME="E5:12_1_1_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</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>, </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:12_1_1_2"/>
for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> holds  1 <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:12_1_1_2_4"/>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<b>thus </b>1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">v</font>
<b>by </b><i/>
<br/>.= 
1 <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T74">RVSUM_1:74</a></i>
;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:12_1_1_2"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</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>

<b>thus </b><a NAME="E3:12_1_1"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_3"/>
for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_3_1"><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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:12_1_1_3_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">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_3_1_1"><b>proof </b></a><div class="add">

<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="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</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> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">w1</font>,<font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<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>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1_1_3"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V3">Abelian</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E4:12_1_1"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V4">add-associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_4"/>
for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:12_1_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font>, <font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E2:12_1_1_4_1_1"/><i><font color="Green">E77</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="euclid.html#K7">+</a> <font color="Maroon">w1</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font></span>)</span>,<font color="Maroon">w1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Maroon">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">u1</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">u1</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</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>
<b>by </b><i>, </i>
;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1_1_4"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V4">add-associative</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/>


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

<b>thus </b><a NAME="E5:12_1_1"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V5">right_zeroed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_5"/>
for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Olive">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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_5_1"><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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:12_1_1_5_1"/>
<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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_5_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E2:12_1_1_5_1_1"/><i><font color="Green">E78</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">zero</font> = <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0 as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T132">FINSEQ_2:132</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<b>thus </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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span></span>)</span> = 
<font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">v2</font> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">zero</font>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T33">RVSUM_1:33</a></i>
;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1_1_5"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V5">right_zeroed</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>;<br/>


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

<a NAME="E6:12_1_1"/>
for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>  ex <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> st <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_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="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:12_1_1_6"/>
 ex <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> st <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_6_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<i><font color="Green">E80</font></i>:  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span> = 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">w</font> =  <a href="euclid.html#K6">-</a> <font color="Maroon">v1</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</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">v2</font> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K6">-</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T40">RVSUM_1:40</a></i>
;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_1_1"/>
 <a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font> is <a href="rlvect_1.html#V6">right_complementable</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>


</div>
