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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">i</font> holds <br/> <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Olive" title="b1">a</font>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b2">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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">  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">i</font> holds <br/> <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">a</font>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b1">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">i</font> implies  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">a</font>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">i</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">  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">a</font>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> </span><br/>

<a NAME="E2:21"/>
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c2">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#D3" target="_self" onmouseover="rs('polynom8/D3')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21"/>
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">a</font>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="polynom8.html#D3" target="_self" onmouseover="rs('polynom8/D3')" onmouseout="rh()">Def2</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/>


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