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

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_1_2"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:30_1_2"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E4:30_1_2"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:30_1_2"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:30_1_2"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="quaterni.html#D7" target="_self" title="QUATERNI:def.7">Def7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">x19</font>, <font color="Maroon" title="c17">x29</font>, <font color="Maroon" title="c18">x39</font>, <font color="Maroon" title="c19">x49</font>, <font color="Maroon" title="c20">y19</font>, <font color="Maroon" title="c21">y29</font>, <font color="Maroon" title="c22">y39</font>, <font color="Maroon" title="c23">y49</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> <span class="kw"> such that </span><br/><a NAME="E8:30_1_2"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<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">x19</font>,<font color="Maroon" title="c17">x29</font>,<font color="Maroon" title="c18">x39</font>,<font color="Maroon" title="c19">x49</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:30_1_2"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
<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">y19</font>,<font color="Maroon" title="c21">y29</font>,<font color="Maroon" title="c22">y39</font>,<font color="Maroon" title="c23">y49</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:30_1_2"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
 <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">x19</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">y19</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">x29</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c21">y29</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">x39</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c22">y39</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">x49</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">y49</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="quaterni.html#D7" target="_self" title="QUATERNI:def.7">Def7</a></span>;<br/>
<a NAME="E11:30_1_2"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E8:30_1_2"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E12:30_1_2"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<font color="Maroon" title="c9">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E8:30_1_2"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E13:30_1_2"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<font color="Maroon" title="c10">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x39</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E8:30_1_2"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E14:30_1_2"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
<font color="Maroon" title="c11">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x49</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E8:30_1_2"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E15:30_1_2"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="quaterni.html#E26"><span class="lab"><font color="Green" title="E20">Lm7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E16:30_1_2"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">y19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E11:30_1_2"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="quaterni.html#E26"><span class="lab"><font color="Green" title="E20">Lm7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E17:30_1_2"/><span class="lab"><font color="Green" title="E36">A19</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="quaterni.html#E26"><span class="lab"><font color="Green" title="E20">Lm7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E18:30_1_2"/><span class="lab"><font color="Green" title="E37">A20</font></span>: 
<font color="Maroon" title="c9">x2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c21">y29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E12:30_1_2"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="quaterni.html#E26"><span class="lab"><font color="Green" title="E20">Lm7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E19:30_1_2"/><span class="lab"><font color="Green" title="E38">A21</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="quaterni.html#E26"><span class="lab"><font color="Green" title="E20">Lm7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E20:30_1_2"/><span class="lab"><font color="Green" title="E39">A22</font></span>: 
<font color="Maroon" title="c10">x3</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c22">y39</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E13:30_1_2"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="quaterni.html#E26"><span class="lab"><font color="Green" title="E20">Lm7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E21:30_1_2"/><span class="lab"><font color="Green" title="E40">A23</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="quaterni.html#E26"><span class="lab"><font color="Green" title="E20">Lm7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E22:30_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">y49</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E14:30_1_2"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="txt" href="quaterni.html#E26"><span class="lab"><font color="Green" title="E20">Lm7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:30_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E9:30_1_2"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="txt" href="#E15:30_1_2"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E16:30_1_2"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E17:30_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E18:30_1_2"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="txt" href="#E19:30_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E20:30_1_2"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="txt" href="#E21:30_1_2"><span class="lab"><font color="Green" title="E40">A23</font></span></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/>


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