<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c4">C1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:93"/><i><font color="Green" title="E52">A1</font></i>: 
for <font color="Olive" title="b1">i</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" title="c3">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>and </b><br/><a NAME="E2:93"/><i><font color="Green" title="E53">A2</font></i>: 
<font color="Maroon" title="c4">C1</font> is    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c3">C</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">NS</font> being  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E4:93"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c4">C1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">NS</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E2:93"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="kurato_2.html#D10" target="_self" title="KURATO_2:def.10">Def10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">A1</font> = <font color="Maroon" title="c1">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">NS</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A1</font> = <font color="Maroon" title="c1">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">NS</font> as   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="kurato_2.html#T42" target="_self" title="KURATO_2:th.42">Th42</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">B1</font> = <font color="Maroon" title="c2">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">NS</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">B1</font> = <font color="Maroon" title="c2">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">NS</font> as   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="kurato_2.html#T42" target="_self" title="KURATO_2:th.42">Th42</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">A1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c9">B1</font>
;<br/>

<a NAME="E7:93"/>
for <font color="Olive" title="b1">i</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" title="c4">C1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <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> ;<br/>

<a NAME="E1:93_1"/><i><font color="Green" title="E55">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">NS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:93_1"/><b>then </b><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c7">A1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">NS</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E3:93_1"/><i><font color="Green" title="E57">A6</font></i>: 
<font color="Maroon" title="c9">B1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">NS</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:93_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<font color="Maroon" title="c4">C1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">i</font> = 
<font color="Maroon" title="c3">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">NS</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
<b>by </b><i><a class="txt" href="kurato_2.html#E4:93"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="kurato_2.html#E1:93_1"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">i</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
<b>by </b><i><a class="txt" href="kurato_2.html#E1:93"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="kurato_2.html#E2:93_1"><i><font color="Green" title="E56">A5</font></i></a>, <a class="txt" href="kurato_2.html#E3:93_1"><i><font color="Green" title="E57">A6</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:93_1"/>
<font color="Maroon" title="c4">C1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c10">i</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:93"/>
( <font color="Maroon" title="c7">A1</font> is    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c9">B1</font> is    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c2">B</font> &amp; ( for <font color="Olive" title="b1">i</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" title="c4">C1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">B1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> ) )
 <b>by </b><i><a class="ref" href="kurato_2.html#D10" target="_self" title="KURATO_2:def.10">Def10</a></i>;<br/>


</div>
