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<span class="kw">let </span><font color="Maroon" title="c11">c1</font>, <font color="Maroon" title="c12">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">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font>, <font color="Olive" title="b4">x4</font>, <font color="Olive" title="b5">y1</font>, <font color="Olive" title="b6">y2</font>, <font color="Olive" title="b7">y3</font>, <font color="Olive" title="b8">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="c1">x</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">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> &amp; <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="b5">y1</font>,<font color="Olive" title="b6">y2</font>,<font color="Olive" title="b7">y3</font>,<font color="Olive" title="b8">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> &amp; <font color="Maroon" title="c11">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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b5">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">y2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b8">y4</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b5">y1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b8">y4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Olive" title="b5">y1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b3">x3</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">y2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b4">x4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b8">y4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b8">y4</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Olive" title="b4">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b5">y1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">y2</font></span>)</span></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> ) &amp;  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font>, <font color="Olive" title="b4">x4</font>, <font color="Olive" title="b5">y1</font>, <font color="Olive" title="b6">y2</font>, <font color="Olive" title="b7">y3</font>, <font color="Olive" title="b8">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="c1">x</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">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> &amp; <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="b5">y1</font>,<font color="Olive" title="b6">y2</font>,<font color="Olive" title="b7">y3</font>,<font color="Olive" title="b8">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> &amp; <font color="Maroon" title="c12">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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b5">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">y2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b8">y4</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b5">y1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b8">y4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Olive" title="b5">y1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b3">x3</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">y2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b4">x4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b8">y4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b8">y4</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Olive" title="b4">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b5">y1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">y2</font></span>)</span></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> ) implies <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">x2</font>, <font color="Maroon" title="c15">x3</font>, <font color="Maroon" title="c16">x4</font>, <font color="Maroon" title="c17">y1</font>, <font color="Maroon" title="c18">y2</font>, <font color="Maroon" title="c19">y3</font>, <font color="Maroon" title="c20">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><a NAME="E2:32_1_2"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c1">x</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">x1</font>,<font color="Maroon" title="c14">x2</font>,<font color="Maroon" title="c15">x3</font>,<font color="Maroon" title="c16">x4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:32_1_2"/><span class="lab"><font color="Green" title="E25">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="c17">y1</font>,<font color="Maroon" title="c18">y2</font>,<font color="Maroon" title="c19">y3</font>,<font color="Maroon" title="c20">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:32_1_2"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c11">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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">y2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">y4</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">y1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">y4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">y3</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c17">y1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">x3</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">y2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">x4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">y4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">y4</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c16">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">y1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">y2</font></span>)</span></span>)</span></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">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font>, <font color="Olive" title="b4">x4</font>, <font color="Olive" title="b5">y1</font>, <font color="Olive" title="b6">y2</font>, <font color="Olive" title="b7">y3</font>, <font color="Olive" title="b8">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="c1">x</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">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b4">x4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> or  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="b5">y1</font>,<font color="Olive" title="b6">y2</font>,<font color="Olive" title="b7">y3</font>,<font color="Olive" title="b8">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> or  not <font color="Maroon" title="c12">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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b5">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">y2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b8">y4</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b5">y1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b8">y4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Olive" title="b5">y1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b3">x3</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">y2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b4">x4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b8">y4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b8">y4</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Olive" title="b4">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b5">y1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">y2</font></span>)</span></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> ) or <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c21">x19</font>, <font color="Maroon" title="c22">x29</font>, <font color="Maroon" title="c23">x39</font>, <font color="Maroon" title="c24">x49</font>, <font color="Maroon" title="c25">y19</font>, <font color="Maroon" title="c26">y29</font>, <font color="Maroon" title="c27">y39</font>, <font color="Maroon" title="c28">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><a NAME="E6:32_1_2"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c1">x</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="c21">x19</font>,<font color="Maroon" title="c22">x29</font>,<font color="Maroon" title="c23">x39</font>,<font color="Maroon" title="c24">x49</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:32_1_2"/><span class="lab"><font color="Green" title="E28">A7</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="c25">y19</font>,<font color="Maroon" title="c26">y29</font>,<font color="Maroon" title="c27">y39</font>,<font color="Maroon" title="c28">y49</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:32_1_2"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c12">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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c21">x19</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c25">y19</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c22">x29</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c26">y29</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">x39</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c27">y39</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">x49</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c28">y49</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c21">x19</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c26">y29</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c22">x29</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c25">y19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">x39</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c28">y49</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">x49</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c27">y39</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c21">x19</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c27">y39</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c25">y19</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c23">x39</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">y29</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c24">x49</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">y49</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c22">x29</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c21">x19</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c28">y49</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c24">x49</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c25">y19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">x29</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c27">y39</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">x39</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c26">y29</font></span>)</span></span>)</span></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="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c2</font></span><br/>

<a NAME="E9:32_1_2"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:32_1_2"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E10:32_1_2"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:32_1_2"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E11:32_1_2"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon" title="c15">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x39</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:32_1_2"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E12:32_1_2"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c16">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x49</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:32_1_2"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E13:32_1_2"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E7:32_1_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E14:32_1_2"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c18">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">y29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E7:32_1_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E15:32_1_2"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<font color="Maroon" title="c19">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">y39</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E7:32_1_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="quaterni.html#T12" target="_self" title="QUATERNI:th.12">Th12</a></span>;<br/>
<a NAME="E16:32_1_2"/>
<font color="Maroon" title="c20">y4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y49</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E7:32_1_2"><span class="lab"><font color="Green" title="E28">A7</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="E17:32_1_2"/>
<font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:32_1_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E9:32_1_2"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E10:32_1_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E11:32_1_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E12:32_1_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E13:32_1_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E14:32_1_2"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E15:32_1_2"><span class="lab"><font color="Green" title="E36">A15</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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