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<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">f</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> implies  ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:57"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E4:57"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> )</span><br/>

<a NAME="E5:57"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<font color="Maroon" title="c4">b</font> <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="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E6:57"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<font color="Maroon" title="c3">a</font> <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="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_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="E1:57_1_1"/>
( <font color="Maroon" title="c3">a</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="c4">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="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">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="57_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:57_1_1_1"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<font color="Maroon" title="c3">a</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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><a NAME="E2:57_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</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="#E1:57"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>;<br/><a NAME="E3:57_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A8</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E1:57_1_1_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="card_4.html#T3" title="CARD_4:th.3">CARD_4:3</a></span>;<br/><a NAME="E4:57_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:57_1_1_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="card_4.html#T3" title="CARD_4:th.3">CARD_4:3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:57_1_1_1"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_1"><span class="lab"><font color="Green" title="E46">A8</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:57_1_1_2"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
<font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><a NAME="E2:57_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</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="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>;<br/><a NAME="E3:57_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A10</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E1:57_1_1_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="card_4.html#T3" title="CARD_4:th.3">CARD_4:3</a></span>;<br/><a NAME="E4:57_1_1_2"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:57_1_1_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="card_4.html#T3" title="CARD_4:th.3">CARD_4:3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:57_1_1_2"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_1_2"><span class="lab"><font color="Green" title="E46">A10</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:57_1_1_3"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:57_1_1_3"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="wsierp_1.html#T31" target="_self" title="WSIERP_1:th.31">Th38</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">d</font></span>)</span> as   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> ;<br/><font color="Maroon" title="c3">a</font> = 
<font color="Maroon" title="c3">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T35" title="PREPOWER:th.35">PREPOWER:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1_3"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="txt" href="#E3:57_1_1_3"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="prepower.html#T43" title="PREPOWER:th.43">PREPOWER:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
<font color="Maroon" title="c7">q</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T8" title="PREPOWER:th.8">PREPOWER:8</a></span>
;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">h</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:57_1_1_3"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T25" target="_self" title="WSIERP_1:th.25">Th32</a></span>;<br/><font color="Maroon" title="c4">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> = 
<font color="Maroon" title="c8">h</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E7:57_1_1_3"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
;<br/><a NAME="E9:57_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="wsierp_1.html#T3" target="_self" title="WSIERP_1:th.3">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:57_1_1_3"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_1_1_3"><span class="lab"><font color="Green" title="E47">A13</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:57"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">f</font> )
 ; <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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