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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</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="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font> holds   <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Olive" title="b1">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>)</span>)</span>,<font color="Olive" title="b3">w</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>)</span>)</span>,<font color="Maroon" title="c4">w</font>)</span><br/>



<span class="lab"><font color="Green" title="E6">A1</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D3" title="VECTSP_1:def.3">VECTSP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D3" title="VECTSP_1:def.3">VECTSP_1:def 3</a></span>
;<br/>
<a NAME="E2:8"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></span>;<br/>
<a NAME="E3:8"/>
<font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></span>;<br/>
<a NAME="E4:8"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="vectsp_2.html#D1" title="VECTSP_2:def.1">VECTSP_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D1" target="_self" title="QUOFIELD:def.1">Def1</a></span>;<br/>
<a NAME="E6:8"/>
<font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T2" target="_self" title="QUOFIELD:th.2">Th2</a></span>;<br/>
<a NAME="E7:8"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="vectsp_2.html#D1" title="VECTSP_2:def.1">VECTSP_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">t</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D1" target="_self" title="QUOFIELD:def.1">Def1</a></span>;<br/>
 <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>)</span>)</span>) = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
;<br/>
<span class="kw">then </span> <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>)</span>)</span>) = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K2" title="QUOFIELD:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="quofield.html#K3" title="QUOFIELD:func.3">`2</a></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
<br/>.= 
 <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>)</span>)</span>,<font color="Maroon" title="c4">w</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:8"/>
 <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c2">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>)</span>)</span>,<font color="Maroon" title="c4">w</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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