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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="midsp_1.html#L1" title="MIDSP_1:struct.1">MidStr</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c1">M</font> is   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>
 ;<br/><b>thus </b><a NAME="E2:44_1"/>
 ex <font color="Olive" title="b1">G</font> being non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a>  ex <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">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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</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 <br/>( <font color="Olive" title="b2">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>,<font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> <font color="Olive" title="b2">w</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c2">M</font> = <font color="Maroon" title="c1">M</font> as   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a> <b>by </b><i><a class="txt" href="midsp_2.html#E1:44_1"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font>;<br/>
<a NAME="E2:44_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
 ex <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</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 <span class="p1">(<span class="default"><a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font></span>)</span> st <br/>( <font color="Olive" title="b1">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="c2">M</font>, <a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font>, <a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> <font color="Olive" title="b1">w</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c2">M</font>
;<br/>

<b>thus </b><a NAME="E1:44_1_1_1"/>
(  <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c2">M</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="c2">M</font>, <a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font>, <a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a>  <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="ref" href="midsp_2.html#T24" target="_self" title="MIDSP_2:th.24">Th24</a>, <a class="ref" href="midsp_2.html#T27" target="_self" title="MIDSP_2:th.27">Th27</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font>
;<br/>

<b>thus </b><a NAME="E3:44_1_1"/>
 ex <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">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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</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 <span class="p1">(<span class="default"><a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font></span>)</span> st <br/>( <font color="Olive" title="b1">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>, <a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c1">M</font>, <a href="midsp_1.html#K18" title="MIDSP_1:func.18">vectgroup</a> <font color="Maroon" title="c2">M</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> <font color="Olive" title="b1">w</font> )
 <b>by </b><i><a class="txt" href="midsp_2.html#E2:44_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c2">G</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> , <font color="Maroon" title="c3">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">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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</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><b> such that </b><a NAME="E2:44"/><i><font color="Green" title="E37">A3</font></i>: 
( <font color="Maroon" title="c3">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>,<font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> <font color="Maroon" title="c3">w</font> )
 ;<br/>

<b>thus </b><a NAME="E3:44"/>
<font color="Maroon" title="c1">M</font> is   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>
 <b>by </b><i><a class="txt" href="midsp_2.html#E2:44"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="midsp_2.html#T23" target="_self" title="MIDSP_2:th.23">Th23</a></i>;<br/>


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