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<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="midsp_2.html#V1" target="_self">midpoint_operator</a>  <a href="rlvect_1.html#L1">LoopStr</a> ;<br/><b>let </b><font color="Maroon">w</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">w</font> <a href="midsp_2.html#R2" target="_self">is_atlas_of</a> <font color="Maroon">S</font>,<font color="Maroon">G</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/>





<b>thus </b><a NAME="E2:38"/>
( <span class="p1">(<span class="default"><a href="midsp_2.html#K6" target="_self">@</a> <font color="Maroon">w</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> implies <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="midsp_2.html#D4" target="_self">Def4</a>, </i>;<br/>

<b>thus </b><a NAME="E3:38"/>
( <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> implies <span class="p1">(<span class="default"><a href="midsp_2.html#K6" target="_self">@</a> <font color="Maroon">w</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>] <b>means </b><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E2:38_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">c'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c'</font>] holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">c'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:38_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>]
 <b>and </b><br/><a NAME="E2:38_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c'</font>]
 ;<br/>

<font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#D4" target="_self">Def4</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c'</font>,<font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#D4" target="_self">Def4</a>, <a class="ref" href="midsp_2.html#T5" target="_self">Th5</a>, <a class="txt" href="midsp_2.html#E1:38_1"><i><font color="Green">E38</font></i></a>, </i>
<br/>.= 
<font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c'</font>,<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="midsp_2.html#D4" target="_self">Def4</a>, </i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#D4" target="_self">Def4</a>, </i>
;<br/>
<a NAME="E4:38_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:38_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#D4" target="_self">Def4</a>, </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c'</font> = <span class="p1">(<span class="default"><a href="midsp_2.html#K6" target="_self">@</a> <font color="Maroon">w</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<a NAME="E3:38_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K6" target="_self">@</a> <font color="Maroon">w</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="midsp_2.html#D4" target="_self">Def4</a>, </i>;<br/>
<b>hence </b><a NAME="E4:38_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K6" target="_self">@</a> <font color="Maroon">w</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="midsp_2.html#T2" target="_self">Th2</a>, <a class="ref" href="midsp_2.html#T4" target="_self">Th4</a></i>;<br/>


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


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