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<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</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>assume </b><a NAME="E1:14"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 ;<br/>

<a NAME="E2:14"/><i><font color="Green" title="E13">A2</font></i>: 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="txt" href="real_ns1.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="real_ns1.html#T4" target="_self" title="REAL_NS1:th.4">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">b1</font> = <font color="Maroon" title="c5">b</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1" title="EUCLID:def.1">EUCLID:def 1</a></i>;<br/>
<font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c3">y</font> = 
<font color="Maroon" title="c4">a</font> <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="c5">b</font></span>)</span>
<b>by </b><i><a class="txt" href="real_ns1.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="real_ns1.html#E2:14"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="real_ns1.html#T2" target="_self" title="REAL_NS1:th.2">Th2</a></i>
<br/>.= 
<font color="Maroon" title="c6">a1</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c7">b1</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52" title="RVSUM_1:th.52">RVSUM_1:52</a></i>
;<br/>
<b>hence </b><a NAME="E5:14"/>
<font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">b</font>
 ;<br/>


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