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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:13_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K1" target="_self">NumberOnLevel</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:13_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K1" target="_self">NumberOnLevel</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:13_1_1_1"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="bintree2.html#K1" target="_self">NumberOnLevel</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bintree2.html#K1" target="_self">NumberOnLevel</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E4:13_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E1:13_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:13_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:13_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E8:13_1_1_1"/><i><font color="Green">E16</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E9:13_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E8:13_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>2</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><a NAME="E11:13_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K2">-level</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E2:13_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E12:13_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D6">TREES_2:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="bintree2.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:13_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E15:13_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E16:13_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E15:13_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>2</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="bintree2.html#K1" target="_self">NumberOnLevel</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="bintree2.html#E4:13_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bintree2.html#E7:13_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="bintree2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="bintree2.html#E3:13_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="bintree2.html#E11:13_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="bintree2.html#E14:13_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="bintree2.html#D1" target="_self">Def1</a></i>
;<br/><a NAME="E19:13_1_1_1"/><b>then </b>
 <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E20:13_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="binari_3.html#T2">BINARI_3:2</a></i>;<br/><b>hence </b><a NAME="E21:13_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="bintree2.html#E7:13_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bintree2.html#E14:13_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="binari_3.html#T3">BINARI_3:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_1_1"/>
 <a href="bintree2.html#K1" target="_self">NumberOnLevel</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


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