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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> ;<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="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ;<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>;<br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E17">A1</font></i>: 
<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>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">b'</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">c'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>









<b>assume </b><a NAME="E2:17"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c8">c'</font> )
 ;<br/>

<b>thus </b><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">c'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b'</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c8">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:17"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c8">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:17"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c8">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:17"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:17"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:17"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="midsp_2.html#T7" target="_self" title="MIDSP_2:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="midsp_2.html#T13" target="_self" title="MIDSP_2:th.13">Th13</a></i>
<br/>.= 
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:17"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</a></i>
;<br/>


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