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<b>let </b><font color="Maroon" title="c6">c1</font>, <font color="Maroon" title="c7">c2</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"> ( <font color="Maroon" title="c1">z</font> <a href="quaterni.html#K7" title="QUATERNI:func.7">+</a> <font color="Maroon" title="c6">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">z</font> <a href="quaterni.html#K7" title="QUATERNI:func.7">+</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c6">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:29_1_2"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c1">z</font> <a href="quaterni.html#K7" title="QUATERNI:func.7">+</a> <font color="Maroon" title="c6">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:29_1_2"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c1">z</font> <a href="quaterni.html#K7" title="QUATERNI:func.7">+</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c2</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font>, <font color="Maroon" title="c10">x3</font>, <font color="Maroon" title="c11">x4</font>, <font color="Maroon" title="c12">y1</font>, <font color="Maroon" title="c13">y2</font>, <font color="Maroon" title="c14">y3</font>, <font color="Maroon" title="c15">y4</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="E4:29_1_2"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c1">z</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">x1</font>,<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font>,<font color="Maroon" title="c11">x4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>and </b><br/><a NAME="E5:29_1_2"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c6">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="c12">y1</font>,<font color="Maroon" title="c13">y2</font>,<font color="Maroon" title="c14">y3</font>,<font color="Maroon" title="c15">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>and </b><br/><a NAME="E6:29_1_2"/><i><font color="Green" title="E24">A6</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c13">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">x3</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">x4</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">y4</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>by </b><i><a class="txt" href="quaterni.html#E1:29_1_2"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="quaterni.html#D7" target="_self" title="QUATERNI:def.7">Def7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">x1'</font>, <font color="Maroon" title="c17">x2'</font>, <font color="Maroon" title="c18">x3'</font>, <font color="Maroon" title="c19">x4'</font>, <font color="Maroon" title="c20">y1'</font>, <font color="Maroon" title="c21">y2'</font>, <font color="Maroon" title="c22">y3'</font>, <font color="Maroon" title="c23">y4'</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="E8:29_1_2"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c1">z</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="c16">x1'</font>,<font color="Maroon" title="c17">x2'</font>,<font color="Maroon" title="c18">x3'</font>,<font color="Maroon" title="c19">x4'</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>and </b><br/><a NAME="E9:29_1_2"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c7">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="c20">y1'</font>,<font color="Maroon" title="c21">y2'</font>,<font color="Maroon" title="c22">y3'</font>,<font color="Maroon" title="c23">y4'</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>and </b><br/><a NAME="E10:29_1_2"/><i><font color="Green" title="E27">A9</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c16">x1'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">y1'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">x2'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c21">y2'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">x3'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c22">y3'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">x4'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">y4'</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>by </b><i><a class="txt" href="quaterni.html#E2:29_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="quaterni.html#D7" target="_self" title="QUATERNI:def.7">Def7</a></i>;<br/>
<a NAME="E11:29_1_2"/><i><font color="Green" title="E28">A10</font></i>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1'</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2'</font> &amp; <font color="Maroon" title="c10">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x3'</font> &amp; <font color="Maroon" title="c11">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x4'</font> )
 
<b>by </b><i><a class="txt" href="quaterni.html#E4:29_1_2"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="quaterni.html#E8:29_1_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></i>;<br/>
<a NAME="E12:29_1_2"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="quaterni.html#E6:29_1_2"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="quaterni.html#E24"><i><font color="Green" title="E18">Lm5</font></i></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></i>;<br/>
<a NAME="E13:29_1_2"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">y1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="quaterni.html#E10:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="quaterni.html#E11:29_1_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="quaterni.html#E24"><i><font color="Green" title="E18">Lm5</font></i></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></i>;<br/>
<a NAME="E14:29_1_2"/><i><font color="Green" title="E31">A13</font></i>: 
<font color="Maroon" title="c9">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c13">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="quaterni.html#E6:29_1_2"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="quaterni.html#E24"><i><font color="Green" title="E18">Lm5</font></i></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></i>;<br/>
<a NAME="E15:29_1_2"/><i><font color="Green" title="E32">A14</font></i>: 
<font color="Maroon" title="c9">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c21">y2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="quaterni.html#E10:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="quaterni.html#E11:29_1_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="quaterni.html#E24"><i><font color="Green" title="E18">Lm5</font></i></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></i>;<br/>
<a NAME="E16:29_1_2"/><i><font color="Green" title="E33">A15</font></i>: 
<font color="Maroon" title="c10">x3</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="quaterni.html#E6:29_1_2"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="quaterni.html#E24"><i><font color="Green" title="E18">Lm5</font></i></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></i>;<br/>
<a NAME="E17:29_1_2"/><i><font color="Green" title="E34">A16</font></i>: 
<font color="Maroon" title="c10">x3</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c22">y3'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="quaterni.html#E10:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="quaterni.html#E11:29_1_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="quaterni.html#E24"><i><font color="Green" title="E18">Lm5</font></i></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></i>;<br/>
<a NAME="E18:29_1_2"/><i><font color="Green" title="E35">A17</font></i>: 
<font color="Maroon" title="c11">x4</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">y4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="quaterni.html#E6:29_1_2"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="quaterni.html#E24"><i><font color="Green" title="E18">Lm5</font></i></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></i>;<br/>
<a NAME="E19:29_1_2"/>
<font color="Maroon" title="c11">x4</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">y4'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="quaterni.html#E10:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="quaterni.html#E11:29_1_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="quaterni.html#E24"><i><font color="Green" title="E18">Lm5</font></i></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E20:29_1_2"/>
<font color="Maroon" title="c6">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c2</font>
 <b>by </b><i><a class="txt" href="quaterni.html#E5:29_1_2"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="quaterni.html#E9:29_1_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="quaterni.html#E12:29_1_2"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="quaterni.html#E13:29_1_2"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="quaterni.html#E14:29_1_2"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="quaterni.html#E15:29_1_2"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="quaterni.html#E16:29_1_2"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="quaterni.html#E17:29_1_2"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="quaterni.html#E18:29_1_2"><i><font color="Green" title="E35">A17</font></i></a></i>; <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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