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<b>let </b><font color="Maroon" title="c1">T</font> be  <a href="bintree2.html#V1" title="BINTREE2:attr.1">full</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> <br/> for <font color="Olive" title="b2">y</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Olive" title="b1">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Olive" title="b1">n</font></span><br/>

<a NAME="E1:25"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<b>by </b><i><a class="ref" href="bintree2.html#D2" target="_self" title="BINTREE2:def.2">Def2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">y</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c2">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c2">n</font></span><br/><b>let </b><font color="Maroon" title="c3">y</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c2">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font> implies <span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c2">n</font> )</span><br/>



<b>assume </b><a NAME="E2:25"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E3:25"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="bintree2.html#T11" target="_self" title="BINTREE2:th.11">Th11</a></i>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">F</font> =  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">y</font></span>)</span> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c2">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E6:25"/><i><font color="Green" title="E21">A4</font></i>: 
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="binari_3.html#T10" title="BINARI_3:th.10">BINARI_3:10</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="#E3:25"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="bintree2.html#D1" target="_self" title="BINTREE2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E6:25"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="binari_3.html#T8" title="BINARI_3:th.8">BINARI_3:8</a></i>
<br/>.= 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c2">n</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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