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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="E1:42"/>
<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/>

<a NAME="E2:42"/><b>then </b>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="midsp_1.html#G1">MidStr</a>(# <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K6" target="_self">@</a> <font color="Maroon">w</font></span>)</span> #) holds <br/> ( <font color="Olive">a</font> <a href="midsp_1.html#K1">@</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> iff <span class="p1">(<span class="default"><a href="midsp_2.html#K7" target="_self">Atlas</a> <font color="Maroon">w</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K7" target="_self">Atlas</a> <font color="Maroon">w</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">c</font>,<font color="Olive">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="E3:42"/>
 <a href="midsp_1.html#G1">MidStr</a>(# <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K6" target="_self">@</a> <font color="Maroon">w</font></span>)</span> #),<font color="Maroon">G</font> <a href="midsp_2.html#R1" target="_self">are_associated_wrp</a>  <a href="midsp_2.html#K7" target="_self">Atlas</a> <font color="Maroon">w</font>
 <b>by </b><i/>;<br/>


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