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<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>





<a NAME="E1:27"/>
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E6"><i><font color="Green" title="E5">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:27"/>
 <a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#D6" target="_self" title="MIDSP_2:def.6">Def6</a></i>;<br/>

 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#T14" target="_self" title="MIDSP_2:th.14">Th14</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#D6" target="_self" title="MIDSP_2:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="ref" href="midsp_2.html#D6" target="_self" title="MIDSP_2:def.6">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E4:27"/>
 <a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="midsp_2.html#D6" target="_self" title="MIDSP_2:def.6">Def6</a></i>;<br/>

<b>thus </b><a NAME="E5:27"/>
(  <a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_2"/>
 <a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c3">y</font>
 ;<br/>

<b>hence </b><font color="Maroon" title="c2">x</font> = 
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#D6" target="_self" title="MIDSP_2:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="ref" href="midsp_2.html#D6" target="_self" title="MIDSP_2:def.6">Def6</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E6:27"/>
 <a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#D6" target="_self" title="MIDSP_2:def.6">Def6</a></i>;<br/>


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