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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>set </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="binom.html#K8" title="BINOM:func.8">In_Power</a> 0;<br/>
<i><font color="Green" title="E22">A1</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="binom.html#K8" title="BINOM:func.8">In_Power</a> 0</span>)</span> = 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="binom.html#D10" target="_self" title="BINOM:def.10">Def10</a></i>
<br/>.= 
1

;<br/>
<a NAME="E2:35"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="binom.html#K8" title="BINOM:func.8">In_Power</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green" title="E24">A3</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="binom.html#K8" title="BINOM:func.8">In_Power</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:35"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="binom.html#K8" title="BINOM:func.8">In_Power</a> 0</span>)</span>
 ;<br/>
<a NAME="E6:35"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="binom.html#E2:35"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="binom.html#E5:35"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">l</font> = 0 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="binom.html#E6:35"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="binom.html#K8" title="BINOM:func.8">In_Power</a> 0</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="binom.html#K8" title="BINOM:func.8">In_Power</a> 0</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="binom.html#E3:35"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="BINOM:func.7">choose</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binom.html#K5" title="BINOM:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c7">l</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E3:35"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="binom.html#E6:35"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="binom.html#D10" target="_self" title="BINOM:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binom.html#K5" title="BINOM:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="binom.html#E6:35"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="binom.html#T8" target="_self" title="BINOM:th.8">Th8</a>, <a class="ref" href="newton.html#T27" title="NEWTON:th.27">NEWTON:27</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binom.html#K5" title="BINOM:func.5">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#T8" target="_self" title="BINOM:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#T14" target="_self" title="BINOM:th.14">Th14</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E10:35"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="binom.html#K8" title="BINOM:func.8">In_Power</a> 0 <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="binom.html#E1:35"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>


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