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<div class="add">

<b>reconsider </b><font color="Maroon">IT</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a>  ;<br/>
<a NAME="E2:49_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font> holds <br/> ( <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">z</font></span>)</span> &amp; <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">IT</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  ex <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font> st <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">IT</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font> ;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y'</font>,<font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font></span>)</span>,<font color="Maroon">z'</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"><font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x'</font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">IT</font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">IT</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">x'</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">R</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>

<b>consider </b><font color="Maroon">b</font> being   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:49_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font> ;<br/>
<b>take </b>
<font color="Maroon">b'</font>
;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b'</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">IT</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a>, <a class="txt" href="vectsp_2.html#E6:49_1_1_1"><i><font color="Green">E21</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:49_1_1"/>
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font> #) is  <a href="rlvect_1.html#V1">strict</a> <a href="vectsp_1.html#NM1">AbGroup</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>;<br/>


</div>
