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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">x</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">n</font>, <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds   <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b3">m</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">n</font>)</span>)</span>,<font color="Olive" title="b3">m</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</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">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds   <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b2">m</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">n</font>)</span>)</span>,<font color="Olive" title="b2">m</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)</span><br/>





 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) = 
<font color="Maroon" title="c2">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</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="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T11" onmouseover="rs('binom/T11')" onmouseout="rh()">BINOM:11</a></span>
<br/>.= 
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c4">m</font>)
<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/>.= 
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</font>)
<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="E2:37"/>
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">n</font>)</span>)</span>,<font color="Maroon" title="c4">m</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"> verum</span><br/>


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