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<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E8">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 ;<br/>

<b>thus </b><a NAME="E2:13"/>
( <font color="Maroon" title="c1">f</font> is <a href="jordan23.html#V2" title="JORDAN23:attr.2">weakly-one-to-one</a> implies <font color="Maroon" title="c1">f</font> is <a href="jordan23.html#V3" title="JORDAN23:attr.3">poorly-one-to-one</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c1">f</font> is <a href="jordan23.html#V2" title="JORDAN23:attr.2">weakly-one-to-one</a>
 ;<br/>

<a NAME="E2:13_1"/><b>then </b>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan23.html#D2" target="_self" title="JORDAN23:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon" title="c1">f</font> is <a href="jordan23.html#V3" title="JORDAN23:attr.3">poorly-one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:13"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="jordan23.html#D3" target="_self" title="JORDAN23:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:13"/>
<font color="Maroon" title="c1">f</font> is <a href="jordan23.html#V3" title="JORDAN23:attr.3">poorly-one-to-one</a>
 ;<br/>

<a NAME="E4:13"/><b>then </b>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:13"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="jordan23.html#D3" target="_self" title="JORDAN23:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E5:13"/>
<font color="Maroon" title="c1">f</font> is <a href="jordan23.html#V2" title="JORDAN23:attr.2">weakly-one-to-one</a>
 <b>by </b><i><a class="ref" href="jordan23.html#D2" target="_self" title="JORDAN23:def.2">Def2</a></i>;<br/>


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