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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">d</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="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">q</font> being  <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font> implies  ex <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:58"/>
 ex <font color="Olive" title="b1">q</font> being  <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</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">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:58"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T31" target="_self" title="WSIERP_1:th.31">Th31</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E36">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_1"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c1">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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font></span><br/><a NAME="E2:58_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">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="#E3:58"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>;<br/><a NAME="E3:58_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">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">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1"><span class="lab"><font color="Green" title="E37">A3</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#T16" title="NEWTON:th.16">NEWTON:16</a></span>;<br/><span class="kw">hence </span><a NAME="E4:58_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</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><span class="lab"><font color="Green" title="E37">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_2"/>
<font color="Maroon" title="c2">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> 
 ; <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">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:58_2_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
( <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">e</font> or <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">e</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="wsierp_1.html#E9"><span class="lab"><font color="Green" title="E7">Lm4</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:58_2_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> or <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T8" title="INT_1:th.8">INT_1:8</a></span>;<br/><span class="kw">assume </span><a NAME="E5:58_2_1"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon" title="c3">k</font> is  not    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font></span><br/><div><span class="lab"><font color="Green" title="E41">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_2_1_1"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">e</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <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">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font></span><br/><a NAME="E2:58_2_1_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">d</font></span>)</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="#E3:58"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E5:58_2_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E4:58_2_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="wsierp_1.html#T3" target="_self" title="WSIERP_1:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_2_1_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E36">A2</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="hs2(this)" href="javascript:()" title="58_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_2_1_2"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">e</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">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font></span><br/><a NAME="E2:58_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:58_2_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="wsierp_1.html#T3" target="_self" title="WSIERP_1:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_2_1_2"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</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><span class="kw">hence </span><a NAME="E8:58_2_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_2_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E6:58_2_1"><span class="lab"><font color="Green" title="E41">A8</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><span class="kw">hence </span><a NAME="E3:58_2"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E35">A1</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="hs2(this)" href="javascript:()" title="58_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_3"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c2">d</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">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font></span><br/><a NAME="E2:58_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>;<br/><a NAME="E3:58_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <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="kw">hence </span><a NAME="E4:58_3"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_3"><span class="lab"><font color="Green" title="E38">A9</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>
<span class="kw">hence </span><a NAME="E7:58"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E37">A4</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/>


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