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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:63"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="rat_1.html#NM1">Rational</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:63"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T31" target="_self">Th31</a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:63_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:63"><i><font color="Green">E45</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E3:63_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E4:63_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:63_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E5:63_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:63_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:63"><i><font color="Green">E45</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E3:63_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E4:63_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:63_2"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_3_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is  not   <a href="nat_1.html#NM1">Nat</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:63_3_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8">INT_1:8</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:63_3_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E14"><i><font color="Green">Lemma9</font></i></a></i>;<br/><div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_3_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:63_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:63"><i><font color="Green">E45</font></i></a>, <a class="txt" href="wsierp_1.html#E3:63_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E3:63_3_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_3_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:63_3_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:63"><i><font color="Green">E45</font></i></a>, <a class="txt" href="wsierp_1.html#E1:63_3_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="wsierp_1.html#E3:63_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E3:63_3_1_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 ;<br/><b>hence </b><a NAME="E4:63_3_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:63"><i><font color="Green">E46</font></i></a>, <a class="ref" href="int_2.html#T8">INT_2:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:63_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:63_3_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="wsierp_1.html#E6:63_3_1"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:63_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:63"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:63"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:63"><i><font color="Green">E47</font></i></a></i>;<br/>


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