<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x3</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/>







<b>thus </b><a NAME="E1:7"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <font color="Maroon">x3</font> implies <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <font color="Maroon">x3</font>
 ;<br/>

<b>thus </b><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x3</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <font color="Maroon">x3</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x3</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">x3</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x3</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x3</font>
 ;<br/><b>thus </b><font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <font color="Maroon">x3</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x3</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x3</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x3</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x3</font> implies <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="euclid.html#K7">+</a> <font color="Maroon">x3</font> )
 ;<br/>


</div>
