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<b>let </b><font color="Maroon" title="c1">c1</font>, <font color="Maroon" title="c2">c2</font>, <font color="Maroon" title="c3">c3</font> be  <a href="quaterni.html#V1" title="QUATERNI:attr.1">quaternion</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">c1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c2">c2</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c3">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c2</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c3">c3</font></span>)</span></span><br/>





<b>consider </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">w1</font>, <font color="Maroon" title="c7">z1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:8"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c7">z1</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>by </b><i><a class="txt" href="quatern2.html#E1"><i><font color="Green" title="E1">QUA7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x2</font>, <font color="Maroon" title="c9">y2</font>, <font color="Maroon" title="c10">w2</font>, <font color="Maroon" title="c11">z2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:8"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c2">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">y2</font>,<font color="Maroon" title="c10">w2</font>,<font color="Maroon" title="c11">z2</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>by </b><i><a class="txt" href="quatern2.html#E1"><i><font color="Green" title="E1">QUA7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">x3</font>, <font color="Maroon" title="c13">y3</font>, <font color="Maroon" title="c14">w3</font>, <font color="Maroon" title="c15">z3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E6:8"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c3">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c12">x3</font>,<font color="Maroon" title="c13">y3</font>,<font color="Maroon" title="c14">w3</font>,<font color="Maroon" title="c15">z3</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>by </b><i><a class="txt" href="quatern2.html#E1"><i><font color="Green" title="E1">QUA7</font></i></a></i>;<br/>
<a NAME="E7:8"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c2">c2</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c3">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">x3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c13">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">w2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">w3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">z2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">z3</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 
<b>by </b><i><a class="txt" href="quatern2.html#E4:8"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="quatern2.html#E6:8"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="quaterni.html#D7" title="QUATERNI:def.7">QUATERNI:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c2">c2</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c3">c3</font> = 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">w1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">w2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">z1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">z2</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c12">x3</font>,<font color="Maroon" title="c13">y3</font>,<font color="Maroon" title="c14">w3</font>,<font color="Maroon" title="c15">z3</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
<b>by </b><i><a class="txt" href="quatern2.html#E2:8"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="quatern2.html#E4:8"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="quatern2.html#E6:8"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="quaterni.html#D7" title="QUATERNI:def.7">QUATERNI:def 7</a></i>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">x2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">x3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c13">y3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">w1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">w2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">w3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">z1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">z2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">z3</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
<b>by </b><i><a class="ref" href="quaterni.html#D7" title="QUATERNI:def.7">QUATERNI:def 7</a></i>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">x3</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c13">y3</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">w1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">w2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">w3</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">z1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">z2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">z3</font></span>)</span></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>

<br/>.= 
<font color="Maroon" title="c1">c1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c2</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c3">c3</font></span>)</span>
<b>by </b><i><a class="txt" href="quatern2.html#E2:8"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="quatern2.html#E7:8"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="quaterni.html#D7" title="QUATERNI:def.7">QUATERNI:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E9:8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c2">c2</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c3">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c2</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c3">c3</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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