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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">X</font> holds <br/> ( <font color="Olive" title="b2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b2">x</font> &amp; (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">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="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   <a href="bcialg_1.html#NM2" title="BCIALG_1:NM.2">BCK-algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">X</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font> )</span><br/>





<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T7" title="BCIALG_1:th.7">BCIALG_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a></span>
;<br/>
<a NAME="E2:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T7" title="BCIALG_1:th.7">BCIALG_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:4"/>
( <font color="Maroon" title="c3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c4">y</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> &amp; (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="bcialg_2.html#K1" title="BCIALG_2:func.1">to_power</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a>, <a class="ref" href="bcialg_2.html#T4" title="BCIALG_2:th.4">BCIALG_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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