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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">x</font>, <font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font>, <font color="Maroon" title="c5">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="E1:8"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font> implies <font color="Maroon" title="c2">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font>
 ;<br/>

<a NAME="E2:8_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1"/>
<font color="Maroon" title="c2">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T11" target="_self" title="EUCLIDLP:th.11">Th11</a></i>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_2"/>
<font color="Maroon" title="c2">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font>
 ;<br/><b>hence </b><font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c3">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T11" target="_self" title="EUCLIDLP:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
( <font color="Maroon" title="c2">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">x2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c5">x3</font> )
 ;<br/>


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