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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>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</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">k</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> <br/>  for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> holds <br/> <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Olive" title="b1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Olive" title="b2">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Olive" title="b1">k</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">k</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">  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>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> holds <br/> <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> implies  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font></span>)</span>) )</span><br/>





<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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">  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font></span>)</span>)</span><br/>

 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c2">k</font>) = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</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="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">k</font></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="#E1:35"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="polynom8.html#E32"><span class="lab"><font color="Green" title="E29">Lm8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font>)</span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 
<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> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="polynom8.html#T18" target="_self" onmouseover="rs('polynom8/T18')" onmouseout="rh()">Th18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:35"/>
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font></span>)</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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