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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" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:13_1_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <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>
 <b>and </b><br/><a NAME="E2:13_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <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>
 <b>and </b><br/><a NAME="E3:13_1_1_1"/><i><font color="Green" title="E13">A3</font></i>: 
<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> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <font color="Maroon" title="c4">x2</font>
 ;<br/><a NAME="E4:13_1_1_1"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c3">x1</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="bintree2.html#E1:13_1_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:13_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="bintree2.html#M1" title="BINTREE2:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D6" title="TREES_2:def.6">TREES_2:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">t1</font> being    <a href="bintree2.html#M1" title="BINTREE2:mode.1">Element</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E7:13_1_1_1"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c5">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font>
 <b>and </b><br/><a NAME="E8:13_1_1_1"/><i><font color="Green" title="E16">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><a NAME="E9:13_1_1_1"/>
 <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> <font color="Maroon" title="c5">t1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="bintree2.html#E8:13_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">F1</font> =  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">t1</font> 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="E11:13_1_1_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c4">x2</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="bintree2.html#E2:13_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E12:13_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="bintree2.html#M1" title="BINTREE2:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_2.html#D6" title="TREES_2:def.6">TREES_2:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">t2</font> being    <a href="bintree2.html#M1" title="BINTREE2:mode.1">Element</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E14:13_1_1_1"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c7">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 <b>and </b><br/><a NAME="E15:13_1_1_1"/><i><font color="Green" title="E19">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><a NAME="E16:13_1_1_1"/>
 <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> <font color="Maroon" title="c7">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="bintree2.html#E15:13_1_1_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">F2</font> =  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c7">t2</font> 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/><span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c6">F1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
<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> <font color="Maroon" title="c3">x1</font>
<b>by </b><i><a class="txt" href="bintree2.html#E4:13_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="bintree2.html#E7:13_1_1_1"><i><font color="Green" title="E15">A5</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"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c8">F2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="bintree2.html#E3:13_1_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="bintree2.html#E11:13_1_1_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="bintree2.html#E14:13_1_1_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="bintree2.html#D1" target="_self" title="BINTREE2:def.1">Def1</a></i>
;<br/><a NAME="E19:13_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F2</font>
 <b>by </b><i><a class="ref" href="binari_3.html#T2" title="BINARI_3:th.2">BINARI_3:2</a></i>;<br/><b>hence </b><a NAME="E20:13_1_1_1"/>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 <b>by </b><i><a class="txt" href="bintree2.html#E7:13_1_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="bintree2.html#E14:13_1_1_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="binari_3.html#T3" title="BINARI_3:th.3">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" title="BINTREE2:func.1">NumberOnLevel</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


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