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<div class="add">

<b>set </b><font color="Maroon" title="c4">D</font> =  <a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>let </b><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font> holds <br/> ex <font color="Olive" title="b2">t</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Olive" title="b2">t</font> ) ) &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font> holds <br/> ex <font color="Olive" title="b2">t</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Olive" title="b2">t</font> ) ) implies <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39_1_2"/><i><font color="Green" title="E23">A8</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font> holds <br/> ex <font color="Olive" title="b2">t</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Olive" title="b2">t</font> ) ) )
 <b>and </b><br/><a NAME="E2:39_1_2"/><i><font color="Green" title="E24">A9</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font> holds <br/> ex <font color="Olive" title="b2">t</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Olive" title="b2">t</font> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font></span><br/>

<a NAME="E3:39_1_2"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font> holds <br/><font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font> implies <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> )</span><br/>

<b>assume </b><a NAME="E1:39_1_2_1"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">t1</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E3:39_1_2_1"/><i><font color="Green" title="E26">A11</font></i>: 
( <font color="Maroon" title="c8">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c8">t1</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E1:39_1_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="osafree.html#E1:39_1_2_1"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">t2</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E5:39_1_2_1"/><i><font color="Green" title="E27">A12</font></i>: 
( <font color="Maroon" title="c9">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c9">t2</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E1:39_1_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="osafree.html#E2:39_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="osafree.html#E1:39_1_2_1"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:39_1_2_1"/>
<font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="osafree.html#E3:39_1_2_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="osafree.html#E5:39_1_2_1"><i><font color="Green" title="E27">A12</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:39_1_2"/>
<font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:39_1_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="osafree.html#E2:39_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
