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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c1">X</font> )
 </span><b>by </b><i><a class="txt" href="fcont_3.html#E1:18"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">X</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:18_1_1_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font> )
 <b>and </b><br/><a NAME="E3:18_1_1_1_1"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c3">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r2</font>
 ;<br/><a NAME="E4:18_1_1_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1_1_1_1"/>
( <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r2</font> or <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r1</font> )
 </span><b>by </b><i><a class="txt" href="fcont_3.html#E3:18_1_1_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r2</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="fcont_3.html#E1:18_1_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="fcont_3.html#E4:18_1_1_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="rfunct_2.html#D2" title="RFUNCT_2:def.2">RFUNCT_2:def 2</a></i>;<br/><a NAME="E3:18_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r1</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font>
 <b>by </b><i><a class="txt" href="fcont_3.html#E1:18_1_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="fcont_3.html#E4:18_1_1_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="rfunct_2.html#D2" title="RFUNCT_2:def.2">RFUNCT_2:def 2</a></i>;<br/><a NAME="E3:18_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font>
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:18_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</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="partfun1.html#T38" title="PARTFUN1:th.38">PARTFUN1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c1">X</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:18_1_1_2_1"/><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font> )
 <b>and </b><br/><a NAME="E3:18_1_1_2_1"/><i><font color="Green" title="E17">A8</font></i>: 
<font color="Maroon" title="c3">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r2</font>
 ;<br/><a NAME="E4:18_1_1_2_1"/><i><font color="Green" title="E18">A9</font></i>: 
( <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_2_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_2_1_1_1"/>
( <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r2</font> or <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r1</font> )
 </span><b>by </b><i><a class="txt" href="fcont_3.html#E3:18_1_1_2_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r2</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font>
 <b>by </b><i><a class="txt" href="fcont_3.html#E1:18_1_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="fcont_3.html#E4:18_1_1_2_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="rfunct_2.html#D3" title="RFUNCT_2:def.3">RFUNCT_2:def 3</a></i>;<br/><a NAME="E3:18_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font>
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_2_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_2_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r1</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="fcont_3.html#E1:18_1_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="fcont_3.html#E4:18_1_1_2_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="rfunct_2.html#D3" title="RFUNCT_2:def.3">RFUNCT_2:def 3</a></i>;<br/><a NAME="E3:18_1_1_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_2_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fcont_3.html#E2:18_1_1_2_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:18_1_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</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="partfun1.html#T38" title="PARTFUN1:th.38">PARTFUN1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>


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