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<b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>set </b><font color="Maroon">S</font> = <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span>;<br/>
<b>consider </b><font color="Maroon">Fr</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:68"/><i><font color="Green">E51</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E3:68"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> 0 <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="binarith.html#K3">-'</a> <font color="Olive">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:68"/><i><font color="Green">E54</font></i>: 
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> 0
 <b>by </b><i><a class="txt" href="catalan2.html#E3:68"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E5:68"/>
0 <a href="hidden.html#R2">in</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
( <font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="binarith.html#K3">-'</a> 0</span>)</span></span>)</span> &amp; 0 <a href="binarith.html#K3">-'</a> 0 <a href="hidden.html#R1">=</a> 0 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, , <a class="ref" href="binarith.html#T51">BINARITH:51</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E7:68"/><b>then </b>
<font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/>
<b>hence </b><a NAME="E8:68"/>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span>
 <b>by </b><i>, <a class="ref" href="stirl2_1.html#T44">STIRL2_1:44</a></i>;<br/>


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