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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">seq1</font>, <font color="Maroon" title="c3">seq2</font>, <font color="Maroon" title="c4">seq3</font> be   <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">N</font>;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c4">seq3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="toprns_1.html#D2" target="_self" title="TOPRNS_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="toprns_1.html#D2" target="_self" title="TOPRNS_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">seq3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c4">seq3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="toprns_1.html#D2" target="_self" title="TOPRNS_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c4">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="toprns_1.html#D2" target="_self" title="TOPRNS_1:def.2">Def2</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c4">seq3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq1</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c4">seq3</font></span>)</span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T6" target="_self" title="TOPRNS_1:th.6">Th6</a></i>;<br/>


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