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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="c1">a</font> &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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">b</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">a9</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c4">b9</font> = <font color="Maroon" title="c2">b</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>  ;<br/>
<span class="kw">thus </span><a NAME="E2:48"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="c1">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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">b</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_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="E1:48_1_1"/>
( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1"/>
<font color="Maroon" title="c2">b</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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="c1">a</font></span><br/><div class="add"><a NAME="E2:48_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_1_1"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D2" title="COMPLEX1:def.2">COMPLEX1:def 2</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="48_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2"/>
<font color="Maroon" title="c2">b</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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="c1">a</font></span><br/><div class="add"><a NAME="E2:48_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 2,<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#T88" title="CARD_1:th.88">CARD_1:88</a></span>;<br/><a NAME="E4:48_1_1_2"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
not <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/><a NAME="E5:48_1_1_2"/>
<font color="Maroon" title="c5">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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_1_1_2"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_2"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="complex1.html#D2" title="COMPLEX1:def.2">COMPLEX1:def 2</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><span class="kw">end;</span></div>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2"><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="E1:48_2"/>
( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1"><span class="kw">suppose </span></a><a NAME="E1:48_2_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c2">b</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">  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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">b</font></span><br/><div class="add"><a NAME="E2:48_2_1"/><span class="kw">then </span>
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_2_1"/>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_2_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="complex1.html#D3" title="COMPLEX1:def.3">COMPLEX1:def 3</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="48_2_2"><span class="kw">suppose </span></a><a NAME="E1:48_2_2"/>
<font color="Maroon" title="c2">b</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">  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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">b</font></span><br/><div class="add"><a NAME="E2:48_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 2,<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#T88" title="CARD_1:th.88">CARD_1:88</a></span>;<br/><a NAME="E4:48_2_2"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
not <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2_2"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/><a NAME="E5:48_2_2"/>
<font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2_2"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_2_2"/>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></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">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_2_2"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="complex1.html#D3" title="COMPLEX1:def.3">COMPLEX1:def 3</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>
