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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">n</font>, <font color="Maroon" title="c6">m</font> be    <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> ;<br/>











<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">m1</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>   st <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Olive" title="b1">m1</font></span>)</span>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Olive" title="b1">m1</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><a NAME="E1:12_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="bcialg_2.html#T1" target="_self" title="BCIALG_2:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E2:12_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="bcialg_2.html#T1" target="_self" title="BCIALG_2:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:12"/><b>then </b><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E3:12"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">k</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">k</font> be    <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> ;<br/><b>assume </b><a NAME="E1:12_2_1"/><i><font color="Green" title="E12">A3</font></i>: 
for <font color="Olive" title="b1">m1</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>   st <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Olive" title="b1">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Olive" title="b1">m1</font></span>)</span>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Olive" title="b1">m1</font>
 ;<br/><b>let </b><font color="Maroon" title="c8">m1</font> be    <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> ;<br/><b>assume </b><a NAME="E2:12_2_1"/><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c8">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c8">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> )
 ;<br/><a NAME="E3:12_2_1"/>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="bcialg_2.html#E2:12_2_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c7">k</font></span>)</span>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="bcialg_2.html#E1:12_2_1"><i><font color="Green" title="E12">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="bcialg_2.html#T4" target="_self" title="BCIALG_2:th.4">Th4</a></i>
;<br/><a NAME="E5:12_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="bcialg_2.html#T7" target="_self" title="BCIALG_2:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E6:12_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="bcialg_2.html#T4" target="_self" title="BCIALG_2:th.4">Th4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_2"/>
for <font color="Olive" title="b1">k</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:12"/>
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="bcialg_2.html#E2:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="bcialg_2.html#E3:12"><i><font color="Green" title="E11">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c3">y</font> <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c5">n</font>
 ;<br/>


</div>
