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

<span class="kw">thus </span><a NAME="E1:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="56_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D4" target="_self" title="QUATERN2:def.4">Def4</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D4" target="_self" title="QUATERN2:def.4">Def4</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T2" target="_self" title="QUATERN2:th.2">Th2</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D4" target="_self" title="QUATERN2:def.4">Def4</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D4" target="_self" title="QUATERN2:def.4">Def4</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="56_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D4" target="_self" title="QUATERN2:def.4">Def4</a></span>
<br/>.= 
<font color="Maroon" title="c1">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T3" target="_self" title="QUATERN2:th.3">Th3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="56_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">y</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E3:56_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T39" target="_self" title="QUATERN2:th.39">Th39</a>, <a class="ref" href="quaterni.html#D8" title="QUATERNI:def.8">QUATERNI:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="56_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D2" title="RLVECT_1:def.2">RLVECT_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="56_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T16" target="_self" title="QUATERN2:th.16">Th16</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E7:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="56_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D7" title="VECTSP_1:def.7">VECTSP_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D4" target="_self" title="QUATERN2:def.4">Def4</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T17" target="_self" title="QUATERN2:th.17">Th17</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D4" target="_self" title="QUATERN2:def.4">Def4</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D4" target="_self" title="QUATERN2:def.4">Def4</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T18" target="_self" title="QUATERN2:th.18">Th18</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D4" target="_self" title="QUATERN2:def.4">Def4</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D6" target="_self" title="QUATERN2:def.6">Def6</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E8:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="56_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:56_1_1_7"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">y</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E4:56_1_1_7"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_7"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">then </span><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
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T31" target="_self" title="QUATERN2:th.31">Th31</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:56_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E9:56_1_1"/>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<a NAME="E10:56_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#D10" target="_self" title="QUATERN2:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:56_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D8" title="STRUCT_0:def.8">STRUCT_0:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
