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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( for <font color="Olive" title="b1">R</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  holds <br/><font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 ) &amp; ( for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 holds <br/>(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 ) ) );<br/>
<b>consider </b><font color="Maroon" title="c1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:12_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:12_1_1"/>
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="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> 
 
<b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">X</font> = <font color="Maroon" title="c1">X</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>take </b><font color="Maroon" title="c3">IT</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">R</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  holds <br/><font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> ) &amp; ( for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> holds <br/>(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> ) ) &amp; ( for <font color="Olive" title="b1">M</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>   st ( for <font color="Olive" title="b2">R</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  holds <br/><font color="Olive" title="b2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> ) &amp; ( for <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">H2</font> &amp; <font color="Olive" title="b2">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> holds <br/>(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">H1</font>,<font color="Olive" title="b3">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">H1</font>,<font color="Olive" title="b3">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> ) ) holds <br/><font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">M</font> ) )</span><br/>

<a NAME="E5:12_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D1" title="ZFMISC_1:def.1">ZFMISC_1:def 1</a></i>;<br/>
<a NAME="E6:12_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c4">A</font> =  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> ;<br/>
<a NAME="E1:12_1_1_1"/>
for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  holds <br/>(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">H1</font>, <font color="Maroon" title="c6">H2</font> be  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H2</font> &amp; <font color="Maroon" title="c5">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  &amp; <font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  implies (  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:12_1_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H2</font>
 <b>and </b><br/><a NAME="E2:12_1_1_1_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c5">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> 
 <b>and </b><br/><a NAME="E3:12_1_1_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>  )</span><br/>

<b>consider </b><font color="Maroon" title="c7">S1</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E5:12_1_1_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c7">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font>
 <b>and </b><br/><a NAME="E6:12_1_1_1_1"/><i><font color="Green" title="E13">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">S1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 <b>by </b><i><a class="txt" href="#E2:12_1_1_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="neckla_2.html#D4" target="_self" title="NECKLA_2:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">S2</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E8:12_1_1_1_1"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c8">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H2</font>
 <b>and </b><br/><a NAME="E9:12_1_1_1_1"/><i><font color="Green" title="E15">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">S2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 <b>by </b><i><a class="txt" href="#E3:12_1_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="neckla_2.html#D4" target="_self" title="NECKLA_2:def.4">Def4</a></i>;<br/>
<a NAME="E10:12_1_1_1_1"/><i><font color="Green" title="E16">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 
<b>by </b><i><a class="txt" href="#E5:12_1_1_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E6:12_1_1_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="#E8:12_1_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="#E9:12_1_1_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="classes2.html#T66" title="CLASSES2:th.66">CLASSES2:66</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">RS</font> =  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c7">S1</font>,<font color="Maroon" title="c8">S2</font> as  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<a NAME="E12:12_1_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">RS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 
<b>by </b><i><a class="txt" href="#E5:12_1_1_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E8:12_1_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="#E10:12_1_1_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="neckla_2.html#D2" target="_self" title="NECKLA_2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E13:12_1_1_1_1"/>
 <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> 
 <b>by </b><i><a class="txt" href="#E5:12_1_1_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E8:12_1_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="neckla_2.html#D4" target="_self" title="NECKLA_2:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> </span><br/>

<b>reconsider </b><font color="Maroon" title="c10">RS'</font> =  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> as  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<a NAME="E15:12_1_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">RS'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 
<b>by </b><i><a class="txt" href="#E10:12_1_1_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="neckla_2.html#D3" target="_self" title="NECKLA_2:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E16:12_1_1_1_1"/>
 <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c6">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> 
 <b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self" title="NECKLA_2:def.4">Def4</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="E2:12_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a> ]
 <b>by </b><i><a class="ref" href="neckla_2.html#D4" target="_self" title="NECKLA_2:def.4">Def4</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="E7:12_1_1"/><b>then </b><i><font color="Green" title="E10">A10</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E5:12_1_1"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<a NAME="E8:12_1_1"/><b>then </b><i><font color="Green" title="E11">A11</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E9:12_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">IT</font>] &amp; ( for <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] holds <br/><font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">IT</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] holds <br/><font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_2_1"/><i><font color="Green" title="E12">A12</font></i>: 
for <font color="Olive" title="b1">R</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  holds <br/><font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">R</font> be  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  implies <font color="Maroon" title="c4">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:12_1_1_2_1_1"/><i><font color="Green" title="E13">A13</font></i>: 
( not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> )
 <b>and </b><br/><a NAME="E2:12_1_1_2_1_1"/><i><font color="Green" title="E14">A14</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font></span><br/>

<a NAME="E3:12_1_1_2_1_1"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Maroon" title="c4">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E1:12_1_1_2_1_1"><i><font color="Green" title="E13">A13</font></i></a>, <a class="txt" href="#E2:12_1_1_2_1_1"><i><font color="Green" title="E14">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:12_1_1_2_1_1"/>
<font color="Maroon" title="c4">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 <b>by </b><i><a class="txt" href="#E7:12_1_1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="#E8:12_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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="E2:12_1_1_2_1"/>
for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> holds <br/>(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">H1</font>, <font color="Maroon" title="c5">H2</font> be  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H2</font> &amp; <font color="Maroon" title="c4">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp; <font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> implies (  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:12_1_1_2_1_2"/><i><font color="Green" title="E13">A15</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H2</font>
 <b>and </b><br/><a NAME="E2:12_1_1_2_1_2"/><i><font color="Green" title="E14">A16</font></i>: 
<font color="Maroon" title="c4">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 <b>and </b><br/><a NAME="E3:12_1_1_2_1_2"/><i><font color="Green" title="E15">A17</font></i>: 
<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> )</span><br/>

<a NAME="E4:12_1_1_2_1_2"/><i><font color="Green" title="E16">A18</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/> <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">Y</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"> ( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:12_1_1_2_1_2_1"/><i><font color="Green" title="E17">A19</font></i>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font></span><br/>

<a NAME="E2:12_1_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E18">A20</font></i>: 
<font color="Maroon" title="c4">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="#E8:12_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="#E2:12_1_1_2_1_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E3:12_1_1_2_1_2_1"/>
<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="#E8:12_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="#E3:12_1_1_2_1_2"><i><font color="Green" title="E15">A17</font></i></a>, <a class="txt" href="#E1:12_1_1_2_1_2_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:12_1_1_2_1_2_1"/>
 <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E1:12_1_1_2_1_2"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="#E1:12_1_1_2_1_2_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="#E2:12_1_1_2_1_2_1"><i><font color="Green" title="E18">A20</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="E5:12_1_1_2_1_2"/>
 <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_2_1_2_2"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/> <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">Y</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"> ( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:12_1_1_2_1_2_2_1"/><i><font color="Green" title="E17">A21</font></i>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font></span><br/>

<a NAME="E2:12_1_1_2_1_2_2_1"/><b>then </b><i><font color="Green" title="E18">A22</font></i>: 
<font color="Maroon" title="c4">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="#E8:12_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="#E2:12_1_1_2_1_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E3:12_1_1_2_1_2_2_1"/>
<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="#E8:12_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="#E3:12_1_1_2_1_2"><i><font color="Green" title="E15">A17</font></i></a>, <a class="txt" href="#E1:12_1_1_2_1_2_2_1"><i><font color="Green" title="E17">A21</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:12_1_1_2_1_2_2_1"/>
 <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E1:12_1_1_2_1_2"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="#E1:12_1_1_2_1_2_2_1"><i><font color="Green" title="E17">A21</font></i></a>, <a class="txt" href="#E2:12_1_1_2_1_2_2_1"><i><font color="Green" title="E18">A22</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="E2:12_1_1_2_1_2_2"/>
 <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font>
 <b>by </b><i><a class="txt" href="#E7:12_1_1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="#E8:12_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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="E6:12_1_1_2_1_2"/>
(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> )
 <b>by </b><i><a class="txt" href="#E7:12_1_1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="#E8:12_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="#E4:12_1_1_2_1_2"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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="E3:12_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">IT</font>]
 <b>by </b><i><a class="txt" href="#E1:12_1_1_2_1"><i><font color="Green" title="E12">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>let </b><font color="Maroon" title="c4">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</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">Y</font>] implies <font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> )</span><br/>

<b>assume </b><a NAME="E2:12_1_1_2"/><i><font color="Green" title="E12">A23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">Y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font></span><br/>

<a NAME="E3:12_1_1_2"/>
<font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )</span><br/>

<b>assume </b><a NAME="E1:12_1_1_2_2"/><i><font color="Green" title="E13">A24</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/>

<a NAME="E2:12_1_1_2_2"/>
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E2:12_1_1_2"><i><font color="Green" title="E12">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:12_1_1_2_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="#E8:12_1_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="#E1:12_1_1_2_2"><i><font color="Green" title="E13">A24</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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="E4:12_1_1_2"/>
<font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 ; <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="E10:12_1_1"/>
( ( for <font color="Olive" title="b1">R</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  holds <br/><font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> ) &amp; ( for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b1">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> holds <br/>(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> ) ) &amp; ( for <font color="Olive" title="b1">M</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="neckla_2.html#K3" title="NECKLA_2:func.3">fin_RelStr</a>   st ( for <font color="Olive" title="b2">R</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  holds <br/><font color="Olive" title="b2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> ) &amp; ( for <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">H2</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">H2</font> &amp; <font color="Olive" title="b2">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> holds <br/>(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b2">H1</font>,<font color="Olive" title="b3">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Olive" title="b2">H1</font>,<font color="Olive" title="b3">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> ) ) holds <br/><font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">M</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
