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<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">M</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="midsp_1.html#L1">MidStr</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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</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>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">w</font> <a href="midsp_2.html#R2" target="_self">is_atlas_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,<font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">M</font>,<font color="Maroon">G</font> <a href="midsp_2.html#R1" target="_self">are_associated_wrp</a> <font color="Maroon">w</font>
 ;<br/>

<a NAME="E3:53"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">b2</font> iff <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b1</font>,<font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="midsp_2.html#T6" target="_self">Th6</a>, <a class="ref" href="midsp_2.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E4:53"/><i><font color="Green">E53</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> <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">b1</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">b1</font>,<font color="Maroon">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>thus </b><a NAME="E5:53"/>
( <font color="Maroon">a</font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">b2</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> <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">b1</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">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="midsp_2.html#T6" target="_self">Th6</a>, <a class="ref" href="midsp_2.html#T8" target="_self">Th8</a>, </i>;<br/>

<b>thus </b><a NAME="E6:53"/>
( <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> <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">b1</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">b2</font></span>)</span> implies <font color="Maroon">a</font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="midsp_2.html#T8" target="_self">Th8</a>, <a class="txt" href="midsp_2.html#E3:53"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>


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