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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> <br/> for <font color="Olive" title="b2">w</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font>  st <font color="Olive" title="b2">w</font> <a href="midsp_2.html#R2" title="MIDSP_2:pred.2">is_atlas_of</a> <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">G</font> holds <br/> <a href="midsp_1.html#G1" title="MIDSP_1:aggr.1">MidStr</a>(# <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K6" title="MIDSP_2:func.6">@</a> <font color="Olive" title="b2">w</font></span>)</span> #),<font color="Olive" title="b1">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a>  <a href="midsp_2.html#K7" title="MIDSP_2:func.7">Atlas</a> <font color="Olive" title="b2">w</font></span><br/><b>let </b><font color="Maroon" title="c2">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">w</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">w</font> <a href="midsp_2.html#R2" title="MIDSP_2:pred.2">is_atlas_of</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">G</font> holds <br/> <a href="midsp_1.html#G1" title="MIDSP_1:aggr.1">MidStr</a>(# <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K6" title="MIDSP_2:func.6">@</a> <font color="Olive" title="b1">w</font></span>)</span> #),<font color="Maroon" title="c2">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a>  <a href="midsp_2.html#K7" title="MIDSP_2:func.7">Atlas</a> <font color="Olive" title="b1">w</font></span><br/><b>let </b><font color="Maroon" title="c3">w</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">w</font> <a href="midsp_2.html#R2" title="MIDSP_2:pred.2">is_atlas_of</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">G</font> implies  <a href="midsp_1.html#G1" title="MIDSP_1:aggr.1">MidStr</a>(# <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K6" title="MIDSP_2:func.6">@</a> <font color="Maroon" title="c3">w</font></span>)</span> #),<font color="Maroon" title="c2">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a>  <a href="midsp_2.html#K7" title="MIDSP_2:func.7">Atlas</a> <font color="Maroon" title="c3">w</font> )</span><br/>





<b>assume </b><a NAME="E1:42"/>
<font color="Maroon" title="c3">w</font> <a href="midsp_2.html#R2" title="MIDSP_2:pred.2">is_atlas_of</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="midsp_1.html#G1" title="MIDSP_1:aggr.1">MidStr</a>(# <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K6" title="MIDSP_2:func.6">@</a> <font color="Maroon" title="c3">w</font></span>)</span> #),<font color="Maroon" title="c2">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a>  <a href="midsp_2.html#K7" title="MIDSP_2:func.7">Atlas</a> <font color="Maroon" title="c3">w</font></span><br/>

<a NAME="E2:42"/><b>then </b>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="midsp_1.html#G1" title="MIDSP_1:aggr.1">MidStr</a>(# <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K6" title="MIDSP_2:func.6">@</a> <font color="Maroon" title="c3">w</font></span>)</span> #) holds <br/> ( <font color="Olive" title="b1">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> iff <span class="p1">(<span class="default"><a href="midsp_2.html#K7" title="MIDSP_2:func.7">Atlas</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K7" title="MIDSP_2:func.7">Atlas</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font> )
 
<b>by </b><i><a class="txt" href="midsp_2.html#E40"><i><font color="Green" title="E32">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:42"/>
 <a href="midsp_1.html#G1" title="MIDSP_1:aggr.1">MidStr</a>(# <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K6" title="MIDSP_2:func.6">@</a> <font color="Maroon" title="c3">w</font></span>)</span> #),<font color="Maroon" title="c2">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a>  <a href="midsp_2.html#K7" title="MIDSP_2:func.7">Atlas</a> <font color="Maroon" title="c3">w</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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