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<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font> be   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">b1</font>, <font color="Maroon" title="c6">b2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>











<b>set </b><font color="Maroon" title="c7">w</font> = the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font>;<br/>
<b>set </b><font color="Maroon" title="c8">G</font> = the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font>;<br/>
<a NAME="E1:55"/><i><font color="Green" title="E38">A1</font></i>: 
the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#R2" title="MIDSP_2:pred.2">is_atlas_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></i>;<br/>
<a NAME="E2:55"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c1">M</font>,the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></i>;<br/>
<a NAME="E3:55"/>
( the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a> &amp; the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp; the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp; the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> &amp; the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> )
 
<b>by </b><i><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E4:55"/>
( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b1</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <font color="Maroon" title="c6">b2</font> iff <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">b2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:55"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:55"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="midsp_2.html#T34" target="_self" title="MIDSP_2:th.34">Th34</a></i>;<br/>


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