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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>(  ex <font color="Olive" title="b1">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b3">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b4">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b3">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or ( $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b3">Aq</font>,<font color="Maroon" title="c2">L</font>  holds <br/>(  not <font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font> or  not $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) ) );<br/>
<a NAME="E1:48_1_1"/><i><font color="Green" title="E36">A1</font></i>: 
for <font color="Olive" title="b1">n</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> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</font> be    <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> ; <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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_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="E1:48_1_1_1_1"/>
(  ex <font color="Olive" title="b1">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b3">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b4">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b3">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or for <font color="Olive" title="b1">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b3">Aq</font>,<font color="Maroon" title="c2">L</font>  holds <br/>(  not <font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font> or  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b3">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b4">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b3">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">A'</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c7">d'</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c2">L</font>, <font color="Maroon" title="c8">Aq</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">dq</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E3:48_1_1_1_1_1"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c7">d'</font>
 <b>and </b><br/><a NAME="E4:48_1_1_1_1_1"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c7">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><b>take </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]</span><br/><b>thus </b><a NAME="E5:48_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 <b>by </b><i><a class="txt" href="lattice8.html#E3:48_1_1_1_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="lattice8.html#E4:48_1_1_1_1_1"><i><font color="Green" title="E38">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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_1_2"/><i><font color="Green" title="E37">A4</font></i>: 
for <font color="Olive" title="b1">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b3">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b3">Aq</font>,<font color="Maroon" title="c2">L</font>  holds <br/>(  not <font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font> or  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><b>take </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]</span><br/><b>thus </b><a NAME="E2:48_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="lattice8.html#E1:48_1_1_1_1_2"><i><font color="Green" title="E37">A4</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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:48_1_1"/><i><font color="Green" title="E37">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E4:48_1_1"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E5:48_1_1"/><i><font color="Green" title="E39">A7</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">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">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="lattice5.html#S1" title="LATTICE5:sch.1">LATTICE5:sch 1</a>(<a class="txt" href="lattice8.html#E1:48_1_1"><i><font color="Green" title="E36">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c4">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( for <font color="Olive" title="b1">n</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>   ex <font color="Olive" title="b2">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b3">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b2">A'</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b4">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b5">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b4">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b4">Aq</font>,<font color="Olive" title="b5">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b2">A'</font>,<font color="Olive" title="b3">d'</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">A'</font>,<font color="Olive" title="b3">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">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">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">Aq</font>,<font color="Olive" title="b5">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) )</span><br/>

<b>thus </b><a NAME="E6:48_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</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="txt" href="lattice8.html#E3:48_1_1"><i><font color="Green" title="E37">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( for <font color="Olive" title="b1">n</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>   ex <font color="Olive" title="b2">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b3">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b2">A'</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b4">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b5">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b4">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b4">Aq</font>,<font color="Olive" title="b5">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b2">A'</font>,<font color="Olive" title="b3">d'</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">A'</font>,<font color="Olive" title="b3">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">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">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">Aq</font>,<font color="Olive" title="b5">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) )</span><br/>

<b>thus </b><a NAME="E7:48_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:48_1_1"><i><font color="Green" title="E38">A6</font></i></a></i>; <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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b3">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b2">A'</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b4">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b5">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b4">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b4">Aq</font>,<font color="Olive" title="b5">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b2">A'</font>,<font color="Olive" title="b3">d'</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">A'</font>,<font color="Olive" title="b3">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">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">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">Aq</font>,<font color="Olive" title="b5">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ] <b>means </b> ex <font color="Olive" title="b1">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b3">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b4">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b3">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> );<br/>
<a NAME="E8:48_1_1"/>
 ex <font color="Olive" title="b1">A'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">A'</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b3">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b4">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b3">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">Aq</font>,<font color="Olive" title="b4">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c1">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d'</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b2">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b3">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b2">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">Aq</font>,<font color="Olive" title="b3">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">d'</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c3">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Aq</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">Aq</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">Aq</font>,<font color="Olive" title="b2">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>consider </b><font color="Maroon" title="c5">q</font> being    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c3">d</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Aq</font> =  <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c3">d</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">dq</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c3">d</font></span>)</span>,<font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E3:48_1_1_2"/><i><font color="Green" title="E40">A8</font></i>: 
<font color="Maroon" title="c7">dq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice8.html#K9" title="LATTICE8:func.9">NextDelta2</a> <font color="Maroon" title="c5">q</font>
 ;<br/>
<b>take </b>
 <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c3">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">dq</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c3">d</font></span>)</span>,<font color="Maroon" title="c2">L</font> st <br/>(  <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c3">d</font>,<font color="Olive" title="b1">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c7">dq</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c7">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>thus </b><a NAME="E4:48_1_1_2"/>
 <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c7">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:48_1_1_2"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="lattice8.html#D11" target="_self" title="LATTICE8:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>thus </b><a NAME="E5:48_1_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:48_1_1"><i><font color="Green" title="E38">A6</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>
<a NAME="E9:48_1_1"/><b>then </b><i><font color="Green" title="E40">A9</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="lattice8.html#E5:48_1_1"><i><font color="Green" title="E39">A7</font></i></a></i>;<br/>
<a NAME="E10:48_1_1"/><i><font color="Green" title="E41">A10</font></i>: 
for <font color="Olive" title="b1">k</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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>given </b><font color="Maroon" title="c6">A'</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c7">d'</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c2">L</font>, <font color="Maroon" title="c8">Aq</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">dq</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c2">L</font><b> such that </b><a NAME="E2:48_1_1_3"/>
<font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c7">d'</font>
 <b>and </b><br/><a NAME="E3:48_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">A'</font>,<font color="Maroon" title="c7">d'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:48_1_1_3"/><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E5:48_1_1_3"/>
 ex <font color="Olive" title="b1">A1</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">d1</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">A1</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b3">AQ</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b4">DQ</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b3">AQ</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">AQ</font>,<font color="Olive" title="b4">DQ</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">d1</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c8">Aq</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d1</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c2">L</font> ex <font color="Olive" title="b2">AQ</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b3">DQ</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b2">AQ</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">AQ</font>,<font color="Olive" title="b3">DQ</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c8">Aq</font>,<font color="Olive" title="b1">d1</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Olive" title="b1">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c9">dq</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">AQ</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">DQ</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">AQ</font>,<font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">AQ</font>,<font color="Olive" title="b2">DQ</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>set </b><font color="Maroon" title="c10">AQ</font> =  <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c9">dq</font>;<br/>
<b>consider </b><font color="Maroon" title="c11">Q</font> being    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c9">dq</font>;<br/>
<b>set </b><font color="Maroon" title="c12">DQ</font> =  <a href="lattice8.html#K9" title="LATTICE8:func.9">NextDelta2</a> <font color="Maroon" title="c11">Q</font>;<br/>
<b>take </b>
 <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c9">dq</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">DQ</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c9">dq</font></span>)</span>,<font color="Maroon" title="c2">L</font> st <br/>(  <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c9">dq</font>,<font color="Olive" title="b1">DQ</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>take </b>
 <a href="lattice8.html#K9" title="LATTICE8:func.9">NextDelta2</a> <font color="Maroon" title="c11">Q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c9">dq</font>, <a href="lattice8.html#K9" title="LATTICE8:func.9">NextDelta2</a> <font color="Maroon" title="c11">Q</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>thus </b><a NAME="E2:48_1_1_3_1"/>
 <a href="lattice8.html#K7" title="LATTICE8:func.7">NextSet2</a> <font color="Maroon" title="c9">dq</font>, <a href="lattice8.html#K9" title="LATTICE8:func.9">NextDelta2</a> <font color="Maroon" title="c11">Q</font> <a href="lattice8.html#R2" title="LATTICE8:pred.2">is_extension2_of</a> <font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D11" target="_self" title="LATTICE8:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>thus </b><a NAME="E3:48_1_1_3_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">Aq</font>,<font color="Maroon" title="c9">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:48_1_1_3"><i><font color="Green" title="E42">A11</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="E6:48_1_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="lattice8.html#E5:48_1_1"><i><font color="Green" title="E39">A7</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>thus </b><a NAME="E11:48_1_1"/>
for <font color="Olive" title="b1">k</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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="lattice8.html#E9:48_1_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="lattice8.html#E10:48_1_1"><i><font color="Green" title="E41">A10</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>
