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

<span class="kw">let </span><font color="Maroon" title="c7">c1</font>, <font color="Maroon" title="c8">c2</font> be    <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"> (  ex <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</font>, <font color="Olive" title="b3">y3</font>, <font color="Olive" title="b4">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>  st <br/>( <font color="Maroon" title="c2">y</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="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b3">y3</font>,<font color="Olive" title="b4">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> &amp; <font color="Maroon" title="c7">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">y1</font></span>)</span>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b3">y3</font>,<font color="Olive" title="b4">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> ) &amp;  ex <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</font>, <font color="Olive" title="b3">y3</font>, <font color="Olive" title="b4">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>  st <br/>( <font color="Maroon" title="c2">y</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="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b3">y3</font>,<font color="Olive" title="b4">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> &amp; <font color="Maroon" title="c8">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">y1</font></span>)</span>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b3">y3</font>,<font color="Olive" title="b4">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> ) implies <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c10">y2</font>, <font color="Maroon" title="c11">y3</font>, <font color="Maroon" title="c12">y4</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><a NAME="E2:72_1_2"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon" title="c2">y</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="c9">y1</font>,<font color="Maroon" title="c10">y2</font>,<font color="Maroon" title="c11">y3</font>,<font color="Maroon" title="c12">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:72_1_2"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c7">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c9">y1</font></span>)</span>,<font color="Maroon" title="c10">y2</font>,<font color="Maroon" title="c11">y3</font>,<font color="Maroon" title="c12">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</font>, <font color="Olive" title="b3">y3</font>, <font color="Olive" title="b4">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>   holds <br/>(  not <font color="Maroon" title="c2">y</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="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b3">y3</font>,<font color="Olive" title="b4">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> or  not <font color="Maroon" title="c8">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">y1</font></span>)</span>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b3">y3</font>,<font color="Olive" title="b4">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> ) or <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c13">y19</font>, <font color="Maroon" title="c14">y29</font>, <font color="Maroon" title="c15">y39</font>, <font color="Maroon" title="c16">y49</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><a NAME="E5:72_1_2"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c2">y</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="c13">y19</font>,<font color="Maroon" title="c14">y29</font>,<font color="Maroon" title="c15">y39</font>,<font color="Maroon" title="c16">y49</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:72_1_2"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<font color="Maroon" title="c8">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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c13">y19</font></span>)</span>,<font color="Maroon" title="c14">y29</font>,<font color="Maroon" title="c15">y39</font>,<font color="Maroon" title="c16">y49</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 ; <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="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c2</font></span><br/>

<a NAME="E7:72_1_2"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_1_2"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E5:72_1_2"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E8:72_1_2"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_1_2"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E5:72_1_2"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E9:72_1_2"/>
<font color="Maroon" title="c11">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y39</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_1_2"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E5:72_1_2"><span class="lab"><font color="Green" title="E49">A4</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="E10:72_1_2"/>
<font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_1_2"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E3:72_1_2"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E5:72_1_2"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E6:72_1_2"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E7:72_1_2"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E8:72_1_2"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</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>
