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<b>reconsider </b><font color="Maroon" title="c1">T'</font> = <font color="Maroon">F<sub>1</sub></font>() as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c2">S'</font> = <font color="Maroon">F<sub>2</sub></font>() as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T'</font> ;<br/>
<a NAME="E3:16_1"/><i><font color="Green" title="E15">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>   ex <font color="Olive" title="b2">m</font> being   <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>  ex <font color="Olive" title="b3">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T'</font> st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S'</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">x</font>] )
 
<b>by </b><i><a class="txt" href="frechet2.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">S1</font> being    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">S'</font><b> such that </b><br/><a NAME="E5:16_1"/><i><font color="Green" title="E16">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">S1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="frechet2.html#S1" target="_self" title="FRECHET2:sch.1">FRECHET2:sch 1</a>(<a class="txt" href="frechet2.html#E3:16_1"><i><font color="Green" title="E15">A2</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">S1</font> = <font color="Maroon" title="c3">S1</font> as    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon" title="c4">S1</font>
; <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   <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>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">S1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>]</span><br/>

<b>thus </b><a NAME="E7:16_1"/>
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">S1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>]
 <b>by </b><i><a class="txt" href="frechet2.html#E5:16_1"><i><font color="Green" title="E16">A3</font></i></a></i>; <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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