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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W</font> being  <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( <font color="Olive" title="b2">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> iff <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">b</font></span>)</span> )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> iff <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</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="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</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>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</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">b</font></span>)</span> )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">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/>
<span class="kw">set </span><font color="Maroon" title="c7">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="E1:62"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
( 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#V3" title="RLVECT_1:attr.3">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#V4" title="RLVECT_1:attr.4">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="algstr_0.html#V13" title="ALGSTR_0:attr.13">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#V2" title="RLVECT_1:attr.2">Abelian</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></span>;<br/>
<a NAME="E2:62"/>
( 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> &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:62"/>
( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</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>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</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">b</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="midsp_2.html#T35" target="_self" title="MIDSP_2:th.35">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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