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

<a NAME="E1:16_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  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="16_1_1_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 <a href="convfun1.html#K4" target="_self">RLS_Real</a> , <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>

<b>reconsider </b><font color="Maroon">vr</font> = <font color="Maroon">v</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">wr</font> = <font color="Maroon">w</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">vr</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">wr</font>
<b>by </b><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>
<a NAME="E2:16_1_1"/><i><font color="Green">E38</font></i>: 
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 <a href="convfun1.html#K4" target="_self">RLS_Real</a>  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="16_1_1_2"><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 <a href="convfun1.html#K4" target="_self">RLS_Real</a> , <font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> , <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>

<b>reconsider </b><font color="Maroon">ur</font> = <font color="Maroon">u</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">vr</font> = <font color="Maroon">v</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">wr</font> = <font color="Maroon">w</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:16_1_1_2"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ur</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">vr</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:16_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<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> <span class="p1">(<span class="default"><font color="Maroon">ur</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">vr</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">wr</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:16_1_1_2"/>
<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">vr</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">wr</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:16_1_1_2"/><b>then </b>
<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> <a href="hidden.html#R1">=</a> <font color="Maroon">ur</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">vr</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">wr</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:16_1_1_2"/>
<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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:16_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  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> <a href="convfun1.html#K4" target="_self">RLS_Real</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_3"><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 <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>

<b>reconsider </b><font color="Maroon">vr</font> = <font color="Maroon">v</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E2:16_1_1_3"/>
 <a href="rlvect_1.html#K1">0.</a> <a href="convfun1.html#K4" target="_self">RLS_Real</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></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> <a href="convfun1.html#K4" target="_self">RLS_Real</a> </span>)</span> = 
<font color="Maroon">vr</font> <a href="binop_2.html#K9">+</a> 0
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>

;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:16_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>   ex <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  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> <a href="convfun1.html#K4" target="_self">RLS_Real</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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 <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>

<b>reconsider </b><font color="Maroon">vr</font> = <font color="Maroon">v</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">w</font> =  <a href="binop_2.html#K7">-</a> <font color="Maroon">vr</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  ;<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">vr</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">vr</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="convfun1.html#K4" target="_self">RLS_Real</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:16_1_1"/><i><font color="Green">E47</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 <a href="convfun1.html#K4" target="_self">RLS_Real</a>  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="16_1_1_5"><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 <a href="convfun1.html#K4" target="_self">RLS_Real</a> , <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>



<b>reconsider </b><font color="Maroon">vr</font> = <font color="Maroon">v</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">wr</font> = <font color="Maroon">w</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:16_1_1_5"/><i><font color="Green">E48</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="binop_2.html#K11">*</a> <font color="Maroon">vr</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="binop_2.html#K11">*</a> <font color="Maroon">wr</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_1_1_5"/>
<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">vr</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">wr</font>
 
<b>by </b><i/>;<br/>
<b>hence </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> = 
<font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">vr</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">wr</font></span>)</span>
<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">vr</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">wr</font></span>)</span>

<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/><br/>


</div><b>end;</b></div>
<a NAME="E6:16_1_1"/><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 <a href="convfun1.html#K4" target="_self">RLS_Real</a>  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="16_1_1_6"><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 <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>



<b>reconsider </b><font color="Maroon">vr</font> = <font color="Maroon">v</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E2:16_1_1_6"/>
( <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="binop_2.html#K11">*</a> <font color="Maroon">vr</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="binop_2.html#K11">*</a> <font color="Maroon">vr</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><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> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">vr</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">vr</font></span>)</span>
<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="binop_2.html#K11">*</a> <font color="Maroon">vr</font>

<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="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E7:16_1_1"/><i><font color="Green">E71</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 <a href="convfun1.html#K4" target="_self">RLS_Real</a>  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="16_1_1_7"><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 <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>



<b>reconsider </b><font color="Maroon">vr</font> = <font color="Maroon">v</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E2:16_1_1_7"/>
<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="binop_2.html#K11">*</a> <font color="Maroon">vr</font>
 
<b>by </b><i/>;<br/>
<b>hence </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">v</font></span>)</span> = 
<font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">vr</font></span>)</span>
<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="binop_2.html#K11">*</a> <font color="Maroon">vr</font>

<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>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E8:16_1_1"/>
for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  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="16_1_1_8"><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 <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>

<b>reconsider </b><font color="Maroon">vr</font> = <font color="Maroon">v</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>thus </b>1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
1 <a href="binop_2.html#K11">*</a> <font color="Maroon">vr</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>

;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:16_1_1"/>
(  <a href="convfun1.html#K4" target="_self">RLS_Real</a>  is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="convfun1.html#K4" target="_self">RLS_Real</a>  is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="convfun1.html#K4" target="_self">RLS_Real</a>  is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="convfun1.html#K4" target="_self">RLS_Real</a>  is <a href="rlvect_1.html#V6">right_complementable</a> &amp;  <a href="convfun1.html#K4" target="_self">RLS_Real</a>  is <a href="rlvect_1.html#V7">RealLinearSpace-like</a> )
 <b>by </b><i>, , , , , , , <a class="ref" href="rlvect_1.html#T7">RLVECT_1:7</a></i>;<br/>


</div>
