<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">k</font>,<font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">k</font> holds <br/><font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c4">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font> implies <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a> )</span><br/>







<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is    <a href="bcialg_5.html#M1" title="BCIALG_5:mode.1">BCI-algebra</a> of <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a></span><br/>

<a NAME="E2:43"/>
 for <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="lab"><font color="Green" title="E24">A2</font></span>: (<span class="p1">(<span class="default">(<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span> = 
(<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T4" onmouseover="rs('bcialg_2/T4')" onmouseout="rh()">BCIALG_2:4</a></span>
<br/>.= 
(<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c5">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="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T7" onmouseover="rs('bcialg_2/T7')" onmouseout="rh()">BCIALG_2:7</a></span>
<br/>.= 
(<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">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="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D5" onmouseover="rs('bcialg_1/D5')" onmouseout="rh()">BCIALG_1:def 5</a></span>
<br/>.= 
(<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">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="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T11" onmouseover="rs('bcialg_2/T11')" onmouseout="rh()">BCIALG_2:11</a></span>
<br/>.= 
(<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">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="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T18" onmouseover="rs('bcialg_2/T18')" onmouseout="rh()">BCIALG_2:18</a></span>
<br/>.= 
(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">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="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T6" onmouseover="rs('bcialg_2/T6')" onmouseout="rh()">BCIALG_2:6</a></span>
<br/>.= 
<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">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="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">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="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T7" onmouseover="rs('bcialg_2/T7')" onmouseout="rh()">BCIALG_2:7</a></span>
<br/>.= 
<span class="p1">(<span class="default">(<span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">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="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T10" onmouseover="rs('bcialg_2/T10')" onmouseout="rh()">BCIALG_2:10</a></span>
<br/>.= 
<span class="p1">(<span class="default">(<span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T10" onmouseover="rs('bcialg_2/T10')" onmouseout="rh()">BCIALG_2:10</a></span>
;<br/>
<a NAME="E2:43_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<span class="p1">(<span class="default">(<span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T21" onmouseover="rs('bcialg_2/T21')" onmouseout="rh()">BCIALG_2:21</a></span>;<br/>
<a NAME="E3:43_1"/>
 <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_5.html#K1" title="BCIALG_5:func.1">Polynom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>,<font color="Maroon" title="c5">y</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="bcialg_5.html#D3" target="_self" onmouseover="rs('bcialg_5/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E4:43_1"/><span class="kw">then </span>
(<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T2" onmouseover="rs('bcialg_1/T2')" onmouseout="rh()">BCIALG_1:2</a></span>;<br/>
<a NAME="E5:43_1"/><span class="kw">then </span>
(<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T2" onmouseover="rs('bcialg_1/T2')" onmouseout="rh()">BCIALG_1:2</a></span>;<br/>
<a NAME="E6:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A4</font></span>: 
(<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T11" onmouseover="rs('bcialg_2/T11')" onmouseout="rh()">BCIALG_2:11</a></span>;<br/>
(<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
(<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T11" onmouseover="rs('bcialg_2/T11')" onmouseout="rh()">BCIALG_2:11</a></span>
<br/>.= 
(<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T18" onmouseover="rs('bcialg_2/T18')" onmouseout="rh()">BCIALG_2:18</a></span>
<br/>.= 
(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T6" onmouseover="rs('bcialg_2/T6')" onmouseout="rh()">BCIALG_2:6</a></span>
<br/>.= 
<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">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="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T7" onmouseover="rs('bcialg_2/T7')" onmouseout="rh()">BCIALG_2:7</a></span>
;<br/>
<a NAME="E8:43_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<a NAME="E9:43_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T11" onmouseover="rs('bcialg_1/T11')" onmouseout="rh()">BCIALG_1:11</a></span>;<br/>
<a NAME="E10:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">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="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
;<br/>
<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">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="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T4" onmouseover="rs('bcialg_2/T4')" onmouseout="rh()">BCIALG_2:4</a></span>
<br/>.= 
<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T7" onmouseover="rs('bcialg_2/T7')" onmouseout="rh()">BCIALG_2:7</a></span>
;<br/>
<a NAME="E12:43_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c5">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="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T21" onmouseover="rs('bcialg_2/T21')" onmouseout="rh()">BCIALG_2:21</a></span>;<br/>
<a NAME="E13:43_1"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="bcialg_5.html#T1" target="_self" onmouseover="rs('bcialg_5/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E14:43_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E15:43_1"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T8" onmouseover="rs('bcialg_1/T8')" onmouseout="rh()">BCIALG_1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:43"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:43"/>
<font color="Maroon" title="c1">X</font> is   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D8" onmouseover="rs('bcialg_1/D8')" onmouseout="rh()">BCIALG_1:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
