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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="midsp_2.html#R1" target="_self">are_associated_wrp</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/>
<a NAME="E2:28"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#D2" target="_self">Def2</a></i>;<br/>
<i><font color="Green">E28</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#T10" target="_self">Th10</a></i>
;<br/>
<a NAME="E4:28"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:28"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#D2" target="_self">Def2</a></i>;<br/>
<i><font color="Green">E31</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#T10" target="_self">Th10</a></i>
;<br/>
 <a href="midsp_2.html#K1" target="_self">Double</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="txt" href="midsp_2.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="txt" href="midsp_2.html#E4:28"><i><font color="Green">E29</font></i></a>, <a class="ref" href="midsp_2.html#T17" target="_self">Th17</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K1" target="_self">Double</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="txt" href="midsp_2.html#E5:28"><i><font color="Green">E30</font></i></a>, <a class="txt" href="midsp_2.html#E6:28"><i><font color="Green">E31</font></i></a>, <a class="ref" href="midsp_2.html#T17" target="_self">Th17</a></i>
<br/>.= 
 <a href="midsp_2.html#K1" target="_self">Double</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="midsp_2.html#K1" target="_self">Double</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#T6" target="_self">Th6</a></i>
;<br/>
<a NAME="E8:28"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T20" target="_self">Th20</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#D3" target="_self">Def3</a></i>
;<br/>
<a NAME="E10:28"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_2.html#E2:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E11:28"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_1.html#K1">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_1.html#K1">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_2.html#D2" target="_self">Def2</a></i>;<br/>


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