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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">x3</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>;<br/>







<b>thus </b><a NAME="E1:7"/>
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x3</font> implies <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">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" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x3</font>
 ;<br/>

<b>thus </b><font color="Maroon" title="c2">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x3</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x3</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c4">x3</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E6"><i><font color="Green" title="E6">Lm5</font></i></a>, <a class="txt" href="euclidlp.html#E1:7_1"><i><font color="Green" title="E13">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c4">x3</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></i>
<br/>.= 
<font color="Maroon" title="c3">x2</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">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" title="E13">A2</font></i>: 
<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x3</font>
 ;<br/><b>thus </b><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x3</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c4">x3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x3</font>
<b>by </b><i><a class="txt" href="euclidlp.html#E6"><i><font color="Green" title="E6">Lm5</font></i></a>, <a class="txt" href="euclidlp.html#E1:7_2"><i><font color="Green" title="E13">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c4">x3</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x3</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></i>
<br/>.= 
<font color="Maroon" title="c2">x1</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7"/>
( <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x3</font> implies <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x3</font> )
 ;<br/>


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