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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be  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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( for <font color="Olive" title="b3">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">F</font> holds  <font color="Olive" title="b3">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">N</font> ) holds <br/> ex <font color="Olive" title="b3">B</font> being   <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">N</font> st <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Olive" title="b2">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( for <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">F</font> holds  <font color="Olive" title="b2">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font> ) holds <br/> ex <font color="Olive" title="b2">B</font> being   <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font> st <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Olive" title="b1">F</font></span><br/>



<span class="kw">defpred </span><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> ] <span class="kw">means </span> ex <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Olive" title="b1">i</font></span>)</span> );<br/>
<span class="kw">let </span><font color="Maroon" title="c3">F</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">F</font> holds  <font color="Olive" title="b1">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font> ) implies  ex <font color="Olive" title="b1">B</font> being   <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font> st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
for <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">F</font> holds  <font color="Olive" title="b1">A</font> is    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font> st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">F</font></span><br/>

<div><span class="lab"><font color="Green" title="E8">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/><span class="kw">assume </span><a NAME="E1:12_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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 <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">x</font> as    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E4:12_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow19.html#D2" target="_self" title="YELLOW19:def.2">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">i</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">y</font>] )</span><br/><span class="kw">thus </span><a NAME="E6:12_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">y</font>]</span><br/><span class="kw">thus </span><a NAME="E7:12_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:12"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
(  <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> <font color="Maroon" title="c3">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> )
 <span class="kw">and </span><br/><a NAME="E5:12"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</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">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c5">B</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></span>;<br/>
<a NAME="E7:12"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">N</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D6" title="WAYBEL_0:def.6">WAYBEL_0:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E9:12"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">N</font>
 <span class="kw">and </span><br/><a NAME="E10:12"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">C</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> as    <a href="yellow19.html#M1" title="YELLOW19:mode.1">Subset</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow19.html#D2" target="_self" title="YELLOW19:def.2">Def2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">F</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<a NAME="E12:12"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon" title="c6">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_9.html#T12" title="WAYBEL_9:th.12">WAYBEL_9:12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:12"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">F</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:12"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:12"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E17:12"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E13">A8</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E13">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E20:12"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">and </span><br/><a NAME="E21:12"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:12"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E12">A7</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_2"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E3:12_2"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">and </span><br/><a NAME="E4:12_2"/><span class="lab"><font color="Green" title="E20">A15</font></span>: 
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12"><span class="lab"><font color="Green" title="E10">A5</font></span></a></span>;<br/><a NAME="E5:12_2"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E3:12_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E6:12_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/><a NAME="E7:12_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:12"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><a NAME="E8:12_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon" title="c13">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:12"><span class="lab"><font color="Green" title="E16">A11</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_9.html#T12" title="WAYBEL_9:th.12">WAYBEL_9:12</a></span>;<br/><font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:12"><span class="lab"><font color="Green" title="E14">A9</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c11">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:12"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="waybel_9.html#T16" title="WAYBEL_9:th.16">WAYBEL_9:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c14">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:12"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="waybel_9.html#T16" title="WAYBEL_9:th.16">WAYBEL_9:16</a></span>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">z</font>

;<br/><span class="kw">hence </span><a NAME="E11:12_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E23:12"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
