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<b>let </b><font color="Maroon">M</font> be   <a href="midsp_1.html#NM1">MidSp</a>;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font>;<br/>
<b>thus </b><a NAME="E1:19"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span> iff <font color="Olive">a</font> is    <a href="midsp_1.html#M1">Vector</a> of <font color="Maroon">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:19_1"/>
( <font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span> iff <font color="Maroon">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13">setvect</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="midsp_1.html#T82">MIDSP_1:82</a></i>;<br/>
<b>hence </b><a NAME="E2:19_1"/>
( <font color="Maroon">a</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span> iff <font color="Maroon">a</font> is    <a href="midsp_1.html#M1">Vector</a> of <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="midsp_1.html#T71">MIDSP_1:71</a></i>;<br/>


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

<b>thus </b> <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span> = 
the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
<br/>.= 
 <a href="midsp_1.html#K17">zerovect</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T85">MIDSP_1:85</a></i>
<br/>.= 
 <a href="midsp_1.html#K9">ID</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D17">MIDSP_1:def 17</a></i>
;<br/>

<b>thus </b><a NAME="E3:19"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="midsp_1.html#M1">Vector</a> of <font color="Maroon">M</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/><font color="Olive">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="midsp_1.html#K10">+</a> <font color="Olive">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="midsp_1.html#M1">Vector</a> of <font color="Maroon">M</font>, <font color="Maroon">y</font> be    <a href="midsp_1.html#M1">Vector</a> of <font color="Maroon">M</font>;<br/>



<b>assume </b><a NAME="E1:19_3"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font>, <font color="Maroon">yy</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13">setvect</a> <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="midsp_1.html#T71">MIDSP_1:71</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18">vectgroup</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_1.html#K15">addvect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">xx</font>,<font color="Maroon">yy</font>
<b>by </b><i>, <a class="ref" href="midsp_1.html#T83">MIDSP_1:83</a></i>
<br/>.= 
<font color="Maroon">xx</font> <a href="midsp_1.html#K14">+</a> <font color="Maroon">yy</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D15">MIDSP_1:def 15</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="midsp_1.html#K10">+</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D14">MIDSP_1:def 14</a></i>
;<br/>


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


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