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<b>thus </b><a NAME="E1:51_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span><br/>

<b>thus </b><a NAME="E2:51_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="quatern2.html#T2" target="_self" title="QUATERN2:th.2">Th5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span><br/>

<b>thus </b><a NAME="E3:51_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c2">x1</font> = <font color="Maroon" title="c1">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a>  <b>by </b><i><a class="ref" href="quatern2.html#D11" target="_self" title="QUATERN2:def.11">Def13</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a> </span>)</span> = 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">x1</font>,<a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a> 
<b>by </b><i><a class="ref" href="quatern2.html#D11" target="_self" title="QUATERN2:def.11">Def13</a></i>
<br/>.= 
<a href="quatern2.html#K10" title="QUATERN2:func.10">addquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">x1</font>,<a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a> 
<b>by </b><i><a class="ref" href="quatern2.html#D11" target="_self" title="QUATERN2:def.11">Def13</a></i>
<br/>.= 
<font color="Maroon" title="c2">x1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a> 
<b>by </b><i><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="quatern2.html#T3" target="_self" title="QUATERN2:th.3">Th6</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E4:51_1_1"/>
 <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c2">x1</font> = <font color="Maroon" title="c1">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a>  <b>by </b><i><a class="ref" href="quatern2.html#D11" target="_self" title="QUATERN2:def.11">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">y</font> =  <a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <font color="Maroon" title="c2">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a>  <b>by </b><i><a class="ref" href="quatern2.html#D11" target="_self" title="QUATERN2:def.11">Def13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">y</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a> </span><br/>

<b>thus </b><a NAME="E3:51_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="quatern2.html#K15" title="QUATERN2:func.15">G_Quaternion</a> 
 <b>by </b><i><a class="ref" href="quatern2.html#T33" target="_self" title="QUATERN2:th.33">Th90</a>, <a class="ref" href="quaterni.html#D8" title="QUATERNI:def.8">QUATERNI:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
