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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="midsp_1.html#L1" title="MIDSP_1:struct.1">MidStr</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">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="c1">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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c1">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> <font color="Olive" title="b2">w</font> holds <br/>for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">M</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b4">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b6">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b5">c</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b6">d</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="midsp_1.html#L1" title="MIDSP_1:struct.1">MidStr</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="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 <font color="Maroon" title="c1">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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> <font color="Olive" title="b1">w</font> holds <br/>for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">d</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b5">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b4">c</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b5">d</font></span>)</span></span><br/><span class="kw">let </span><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">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 <font color="Maroon" title="c1">G</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">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>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> <font color="Maroon" title="c3">w</font> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> holds  <span class="p1">(<span class="default"><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></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b4">d</font></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<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="c2">M</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">G</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a> <font color="Maroon" title="c3">w</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">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> holds  <span class="p1">(<span class="default"><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></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b4">d</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">d</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span><br/>

<span class="lab"><font color="Green" title="E26">A3</font></span>: <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>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> = 
<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>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c4">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#T10" target="_self" title="MIDSP_2:th.10">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span>,<font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#T10" target="_self" title="MIDSP_2:th.10">Th10</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">p</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>;<br/>
<a NAME="E4:31"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>,<font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">Def2</a></span>;<br/>
<a NAME="E5:31"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<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>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>,<font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">Def2</a></span>;<br/>
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c6">c</font></span>)</span> = 
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c4">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#T10" target="_self" title="MIDSP_2:th.10">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c6">c</font></span>)</span>,<font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#T10" target="_self" title="MIDSP_2:th.10">Th10</a></span>
;<br/>
<span class="kw">then </span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c6">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span> = 
<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="c7">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="midsp_2.html#T17" target="_self" title="MIDSP_2:th.17">Th17</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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="c7">d</font>,<font color="Maroon" title="c4">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="midsp_2.html#T6" target="_self" title="MIDSP_2:th.6">Th6</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="midsp_2.html#T17" target="_self" title="MIDSP_2:th.17">Th17</a></span>
<br/>.= 
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T15" target="_self" title="MIDSP_2:th.15">Th15</a></span>
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="midsp_2.html#T6" target="_self" title="MIDSP_2:th.6">Th6</a></span>
;<br/>
<a NAME="E8:31"/><span class="kw">then </span>
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c6">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T20" target="_self" title="MIDSP_2:th.20">Th20</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E29">A6</font></span>: <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c6">c</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</a></span>
<br/>.= 
<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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</a></span>
;<br/>
<a NAME="E10:31"/>
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">Def2</a></span>;<br/>
<a NAME="E11:31"/><span class="kw">then </span>
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c6">c</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:31"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c7">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">Def2</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/>


</div>
