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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="group_1.html#V2">unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K8" target="_self">In_Power</a> 0;<br/>
<i><font color="Green">E26</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K8" target="_self">In_Power</a> 0</span>)</span> = 
0 <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="binom.html#D10" target="_self">Def10</a></i>
<br/>.= 
1

;<br/>
<a NAME="E2:35"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K8" target="_self">In_Power</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green">E28</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K8" target="_self">In_Power</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:35"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K8" target="_self">In_Power</a> 0</span>)</span>
 ;<br/>
<a NAME="E6:35"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="binom.html#E2:35"><i><font color="Green">E27</font></i></a>, <a class="txt" href="binom.html#E5:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = 0 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="binom.html#E6:35"><i><font color="Green">E30</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K8" target="_self">In_Power</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K8" target="_self">In_Power</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="binom.html#E3:35"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="binom.html#K7" target="_self">choose</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binom.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E3:35"><i><font color="Green">E28</font></i></a>, <a class="txt" href="binom.html#E6:35"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binom.html#D10" target="_self">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binom.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binom.html#K2" target="_self">|^</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E6:35"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binom.html#T8" target="_self">Th8</a>, <a class="ref" href="newton.html#T27">NEWTON:27</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binom.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#T14" target="_self">Th14</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E10:35"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binom.html#K8" target="_self">In_Power</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="binom.html#E1:35"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>


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