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<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">z</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c2">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> holds <br/> ex <font color="Olive" title="b4">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font>, <font color="Maroon" title="c5">z</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> implies  ex <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:144"/><span class="lab"><font color="Green" title="E92">A1</font></span>: 
<font color="Maroon" title="c2">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  or  not <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> or  ex <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:144"/>
<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> or  ex <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> )</span><br/>

<a NAME="E3:144"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A2</font></span>: 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c4">j</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T34" onmouseover="rs('int_2/T34')" onmouseout="rh()">INT_2:34</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:144"/><span class="lab"><font color="Green" title="E94">A3</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E5:144"/><span class="lab"><font color="Green" title="E95">A4</font></span>: 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c4">j</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>;<br/>
<a NAME="E6:144"/>
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="taylor_2.html#T1" onmouseover="rs('taylor_2/T1')" onmouseout="rh()">TAYLOR_2:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:144"/><span class="lab"><font color="Green" title="E96">A5</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:144"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="txt" href="#E4:144"><span class="lab"><font color="Green" title="E94">A3</font></span></a>, <a class="txt" href="#E5:144"><span class="lab"><font color="Green" title="E95">A4</font></span></a>, <a class="ref" href="newton02.html#T30" onmouseover="rs('newton02/T30')" onmouseout="rh()">NEWTON02:30</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="144_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:144_1"/>
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="144_1_1"><span class="kw">suppose </span></a><a NAME="E1:144_1_1"/><span class="lab"><font color="Green" title="E97">A6</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c6">k</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><a NAME="E2:144_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:144"><span class="lab"><font color="Green" title="E96">A5</font></span></a>, <a class="txt" href="#E1:144_1_1"><span class="lab"><font color="Green" title="E97">A6</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="144_1_2"><span class="kw">suppose </span></a><a NAME="E1:144_1_2"/>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span><br/><div class="add"><a NAME="E2:144_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A7</font></span>: 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T30" onmouseover="rs('absvalue/T30')" onmouseout="rh()">ABSVALUE:30</a></span>;<br/><span class="kw">take </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c6">k</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:144"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="ref" href="power.html#T2" onmouseover="rs('power/T2')" onmouseout="rh()">POWER:2</a></span>
<br/>.= 
<font color="Maroon" title="c3">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:144"><span class="lab"><font color="Green" title="E96">A5</font></span></a>, <a class="txt" href="#E2:144_1_2"><span class="lab"><font color="Green" title="E97">A7</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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