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

<span class="kw">reconsider </span><font color="Maroon" title="c5">x9</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c6">y9</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c7">w9</font> = <font color="Maroon" title="c3">w</font>, <font color="Maroon" title="c8">z9</font> = <font color="Maroon" title="c4">z</font> as    <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">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:17_1_1"/>
( <font color="Maroon" title="c3">w</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="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a>  ex <font color="Olive" title="b2">x9</font>, <font color="Olive" title="b3">y9</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="Olive" title="b2">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b3">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Olive" title="b2">x9</font>,<font color="Olive" title="b3">y9</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  not <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a>  st <font color="Olive">b<sub>1</sub></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#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17_1_1_1"/>
<font color="Maroon" title="c3">w</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="E2:17_1_1_1"/>
<font color="Maroon" title="c4">z</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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a>  ex <font color="Olive" title="b2">x9</font>, <font color="Olive" title="b3">y9</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="Olive" title="b2">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b3">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Olive" title="b2">x9</font>,<font color="Olive" title="b3">y9</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">t</font> = <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">x9</font>,<font color="Maroon" title="c6">y9</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">t</font> = <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">x9</font>,<font color="Maroon" title="c6">y9</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">t</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</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="Olive" title="b1">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Olive" title="b1">x9</font>,<font color="Olive" title="b2">y9</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">x9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y9</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="c5">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">x9</font>,<font color="Olive" title="b1">y9</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">y9</font>
; <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="c5">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c6">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">x9</font>,<font color="Maroon" title="c6">y9</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E4:17_1_1_1"/>
( <font color="Maroon" title="c5">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c6">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">x9</font>,<font color="Maroon" title="c6">y9</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</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>

<span class="kw">set </span><font color="Maroon" title="c9">e</font> = <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="c5">x9</font>,<font color="Maroon" title="c6">y9</font>,<font color="Maroon" title="c7">w9</font>,<font color="Maroon" title="c8">z9</font>;<br/>
<span class="kw">thus </span><a NAME="E3:17_1_1"/>
( ( <font color="Maroon" title="c3">w</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="c4">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a>  st <font color="Olive" title="b1">t</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#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:17_1_1_2"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
( <font color="Maroon" title="c3">w</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="c4">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a>  st <font color="Olive" title="b1">t</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#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font></span><br/>

<a NAME="E2:17_1_1_2"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<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="c5">x9</font>,<font color="Maroon" title="c6">y9</font>,<font color="Maroon" title="c7">w9</font>,<font color="Maroon" title="c8">z9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_1_1_2_1"/>
<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="c5">x9</font>,<font color="Maroon" title="c6">y9</font>,<font color="Maroon" title="c7">w9</font>,<font color="Maroon" title="c8">z9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</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">r</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">r</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:17_1_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</font> being   <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>  st <br/>( <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="c5">x9</font>,<font color="Maroon" title="c6">y9</font>,<font color="Maroon" title="c7">w9</font>,<font color="Maroon" title="c8">z9</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</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">r</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>  )
 ;<br/><span class="kw">hence </span><a NAME="E3:17_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1_2"><span class="lab"><font color="Green" title="E8">A1</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>; <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>
<a NAME="E4:17_1_1_2"/><span class="kw">then </span>
<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="c5">x9</font>,<font color="Maroon" title="c6">y9</font>,<font color="Maroon" title="c7">w9</font>,<font color="Maroon" title="c8">z9</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">r</font> where <font color="Olive" title="b1">r</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">r</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">r</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="#E2:17_1_1_2"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <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="c5">x9</font>,<font color="Maroon" title="c6">y9</font>,<font color="Maroon" title="c7">w9</font>,<font color="Maroon" title="c8">z9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">e</font>
; <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="c10">e</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#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font></span><br/>

<span class="kw">thus </span><a NAME="E6:17_1_1_2"/>
<font color="Maroon" title="c10">e</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#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>
 ; <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>
