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<span class="kw">let </span><font color="Maroon" title="c1">z</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> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> implies <font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c1">z</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">x</font> = <font color="Maroon" title="c1">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:8"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="quatern3.html#E2"><span class="lab"><font color="Green" title="E2">Lm1</font></span></a></span>;<br/>
<font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a> = 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T24" title="QUATERNI:th.24">QUATERNI:24</a></span>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></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="#E1:8"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="quaterni.html#T35" title="QUATERNI:th.35">QUATERNI:35</a></span>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></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="#E1:8"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="quaterni.html#T35" title="QUATERNI:th.35">QUATERNI:35</a></span>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a></span>)</span></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="#E1:8"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="quaterni.html#T35" title="QUATERNI:th.35">QUATERNI:35</a></span>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</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="#E3:8"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="quaterni.html#T35" title="QUATERNI:th.35">QUATERNI:35</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:8"/>
<font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font></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"> verum</span><br/>


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