<?xml version="1.0"?>
<div class="add">

<b>thus </b><a NAME="E1:59_1_1"/>
 <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V34" title="ALGSTR_0:attr.34">almost_right_invertible</a> &amp; not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a> ; <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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">y1</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c6">z1</font> = <font color="Maroon" title="c3">z</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#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font> = 
<a href="quatern2.html#K10" title="QUATERN2:func.10">addquaternion</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<font color="Maroon" title="c6">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</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> <span class="p1">(<span class="default"><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="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<font color="Maroon" title="c6">z1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<font color="Maroon" title="c6">z1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c5">y1</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c6">z1</font>
<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="c4">x1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#T2" target="_self" title="QUATERN2:th.2">Th5</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="c4">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></i>
<br/>.= 
<a href="quatern2.html#K10" title="QUATERN2:func.10">addquaternion</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><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="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></i>
<br/>.= 
<a href="quatern2.html#K10" title="QUATERN2:func.10">addquaternion</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<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"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</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="E2:59_1_1"/>
 <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V34" title="ALGSTR_0:attr.34">almost_right_invertible</a> &amp; not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_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#K16" title="QUATERN2:func.16">R_Quaternion</a> ; <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"> <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#K16" title="QUATERN2:func.16">R_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#D12" target="_self" title="QUATERN2:def.12">Def11</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#K16" title="QUATERN2:func.16">R_Quaternion</a> </span>)</span> = 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_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#D12" target="_self" title="QUATERN2:def.12">Def11</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#D12" target="_self" title="QUATERN2:def.12">Def11</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="E3:59_1_1"/>
 <a href="quatern2.html#K16" title="QUATERN2:func.16">R_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"> (  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V34" title="ALGSTR_0:attr.34">almost_right_invertible</a> &amp; not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_3"><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#K16" title="QUATERN2:func.16">R_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#D12" target="_self" title="QUATERN2:def.12">Def11</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#K16" title="QUATERN2:func.16">R_Quaternion</a>  <b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</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#K16" title="QUATERN2:func.16">R_Quaternion</a> </span><br/>

<b>thus </b><a NAME="E3:59_1_1_3"/>
<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#K16" title="QUATERN2:func.16">R_Quaternion</a> 
 <b>by </b><i><a class="ref" href="quatern2.html#T37" target="_self" title="QUATERN2:th.37">Th7</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>

<b>thus </b><a NAME="E4:59_1_1"/>
 <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V34" title="ALGSTR_0:attr.34">almost_right_invertible</a> &amp; not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a> ; <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"> <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c1">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c3">x1</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c4">y1</font> = <font color="Maroon" title="c2">y</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#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>;<br/>
<b>thus </b><a NAME="E2:59_1_1_4"/>
<font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c1">x</font>
 ; <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="E5:59_1_1"/>
 <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V34" title="ALGSTR_0:attr.34">almost_right_invertible</a> &amp; not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">y1</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c6">z1</font> = <font color="Maroon" title="c3">z</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#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font> = 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<font color="Maroon" title="c6">z1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<font color="Maroon" title="c6">z1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<font color="Maroon" title="c6">z1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c5">y1</font></span>)</span> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c6">z1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#T16" target="_self" title="QUATERN2:th.16">Th9</a></i>
<br/>.= 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</a></i>
<br/>.= 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</a></i>
<br/>.= 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x1</font>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</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="E6:59_1_1"/>
(  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V34" title="ALGSTR_0:attr.34">almost_right_invertible</a> &amp; not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><br/>

<b>thus </b><a NAME="E7:59_1_1"/>
 <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V34" title="ALGSTR_0:attr.34">almost_right_invertible</a> &amp; not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</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#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">y1</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c6">z1</font> = <font color="Maroon" title="c3">z</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#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> = 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x1</font>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<span class="p1">(<span class="default"><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="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c6">z1</font></span>)</span>
<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="c4">x1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c5">y1</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#T17" target="_self" title="QUATERN2:th.17">Th10</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</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> <span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</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> <span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span></span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c1">x</font> = 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span>)</span>,<font color="Maroon" title="c4">x1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><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="c5">y1</font>,<font color="Maroon" title="c6">z1</font></span>)</span>,<font color="Maroon" title="c4">x1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c6">z1</font></span>)</span>,<font color="Maroon" title="c4">x1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c6">z1</font></span>)</span> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c4">x1</font>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#T18" target="_self" title="QUATERN2:th.18">Th11</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">z1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</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> <span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">z1</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</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> <span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">z1</font>,<font color="Maroon" title="c4">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D8" target="_self" title="QUATERN2:def.8">Def8</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">z1</font>,<font color="Maroon" title="c4">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quatern2.html#K12" title="QUATERN2:func.12">multquaternion</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">z1</font>,<font color="Maroon" title="c4">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c1">x</font></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</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="E8:59_1_1"/>
 <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="algstr_0.html#V34" title="ALGSTR_0:attr.34">almost_right_invertible</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_7"><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#K16" title="QUATERN2:func.16">R_Quaternion</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D40" title="ALGSTR_0:def.40">ALGSTR_0:def 40</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="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#K16" title="QUATERN2:func.16">R_Quaternion</a>  or <font color="Maroon" title="c1">x</font> is <a href="algstr_0.html#V24" title="ALGSTR_0:attr.24">right_invertible</a> )</span><br/>

<b>assume </b><a NAME="E1:59_1_1_7"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <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> <a href="quatern2.html#K16" title="QUATERN2:func.16">R_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> is <a href="algstr_0.html#V24" title="ALGSTR_0:attr.24">right_invertible</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#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">y</font> = <font color="Maroon" title="c2">x1</font> <a href="quatern2.html#K8" title="QUATERN2:func.8">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</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#D28" title="ALGSTR_0:def.28">ALGSTR_0:def 28</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#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a> </span><br/>

<a NAME="E4:59_1_1_7"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a> 
 
<b>by </b><i><a class="txt" href="quatern2.html#E1:59_1_1_7"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>;<br/>
<font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font> = 
1
<b>by </b><i><a class="txt" href="quatern2.html#E4:59_1_1_7"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="quatern2.html#T29" target="_self" title="QUATERN2:th.29">Thaa</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a> 
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>
;<br/>
<b>hence </b><a NAME="E6:59_1_1_7"/>
<font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a> 
 ; <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="E9:59_1_1"/>
not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</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="59_1_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:59_1_1_8"/><i><font color="Green" title="E35">A</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quatern2.html#K2" title="QUATERN2:func.2">1q</a> 
 
<b>by </b><i><a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>;<br/>
<a NAME="E2:59_1_1_8"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a> 
 
<b>by </b><i><a class="txt" href="quatern2.html#E1:59_1_1_8"><i><font color="Green" title="E35">A</font></i></a>, <a class="ref" href="quatern2.html#D12" target="_self" title="QUATERN2:def.12">Def11</a></i>;<br/>
<b>hence </b><a NAME="E3:59_1_1_8"/>
not  <a href="quatern2.html#K16" title="QUATERN2:func.16">R_Quaternion</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D8" title="STRUCT_0:def.8">STRUCT_0: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>
