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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>assume </b><a NAME="E1:26"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:26"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i1</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="newton.html#K10">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#D4">NEWTON:def 4</a></i>;<br/>
<a NAME="E6:26"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:26"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="newton.html#K10">In_Power</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="newton.html#K10">In_Power</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="newton.html#K2">|^</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="newton.html#K2">|^</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#D4">NEWTON:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="irrat_1.html#NK1" target="_self">^</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="newton.html#K2">|^</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T48">POWER:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K9">choose</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="irrat_1.html#NK1" target="_self">^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="irrat_1.html#NK1" target="_self">^</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T48">POWER:48</a></i>
;<br/><br/>


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