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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="Red">::  thesis: </font></span></a><span class="hide"> <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"><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 class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span></span><br/>

<a NAME="E1:51"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#D2" target="_self" title="QUATERNI:def.2">Def2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1"/>
( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  or  not <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_1_1"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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"><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 class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span></span><br/><div class="add"><a NAME="E2:51_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A3</font></span>: 
 <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</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="ref" href="quaterni.html#D15" target="_self" title="QUATERNI:def.15">Def15</a></span>;<br/><a NAME="E3:51_1_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
 <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z</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="#E1:51_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="quaterni.html#D16" target="_self" title="QUATERNI:def.16">Def16</a></span>;<br/><a NAME="E4:51_1_1"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
 ex <font color="Olive" title="b1">z9</font> being  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z9</font> &amp;  <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">z9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="quaterni.html#D13" target="_self" title="QUATERNI:def.13">Def13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c2">z9</font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E6:51_1_1"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z9</font>
 <span class="kw">and </span><br/><a NAME="E7:51_1_1"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
 <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="quaterni.html#D14" target="_self" title="QUATERNI:def.14">Def14</a></span>;<br/><a NAME="E8:51_1_1"/>
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><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 class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E6:51_1_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E7:51_1_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="quaterni.html#E47"><span class="lab"><font color="Green" title="E29">Lm9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:51_1_1"/>
<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"><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 class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</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="#E2:51_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:51_1_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:51_1_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="quaterni.html#E15"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_1_2"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
not <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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"><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 class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span></span><br/><div class="add"><a NAME="E2:51_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#D14" target="_self" title="QUATERNI:def.14">Def14</a></span>;<br/><a NAME="E3:51_1_2"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.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="#E1:51_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="quaterni.html#D13" target="_self" title="QUATERNI:def.13">Def13</a></span>;<br/><a NAME="E4:51_1_2"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="quaterni.html#D15" target="_self" title="QUATERNI:def.15">Def15</a></span>;<br/><a NAME="E5:51_1_2"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="quaterni.html#D16" target="_self" title="QUATERNI:def.16">Def16</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</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="E7:51_1_2"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2,3 <a href="quaterni.html#K3" title="QUATERNI:func.3">--&gt;</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="quaterni.html#T22" target="_self" title="QUATERNI:th.22">Th22</a></span>;<br/><a NAME="E8:51_1_2"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
 <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E7:51_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="quaterni.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>;<br/><a NAME="E9:51_1_2"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
 <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E7:51_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="quaterni.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>;<br/><a NAME="E10:51_1_2"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
 <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_2"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E7:51_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="quaterni.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>;<br/><a NAME="E11:51_1_2"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
 <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1_2"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E7:51_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="quaterni.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>;<br/><a NAME="E12:51_1_2"/>
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : ( <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E1:51_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E13:51_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A18</font></span>: 
not <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : ( <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c1">z</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="card_1.html#T90" title="CARD_1:th.90">CARD_1:90</a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/><a NAME="E15:51_1_2"/>
( <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51_1_2"><span class="lab"><font color="Green" title="E43">A18</font></span></a></span>;<br/><a NAME="E16:51_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="quaterni.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E17:51_1_2"/>
<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"><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 class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</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="#E7:51_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E8:51_1_2"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="txt" href="#E9:51_1_2"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="txt" href="#E10:51_1_2"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="txt" href="#E11:51_1_2"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="quaterni.html#D6" target="_self" title="QUATERNI:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
