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

<span class="kw">let </span><font color="Maroon" title="c1">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"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st ( for <font color="Olive" title="b4">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">D</font> iff  ex <font color="Olive" title="b5">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font> &amp;  ex <font color="Olive" title="b6">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Olive" title="b2">M</font> st <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b6">C</font> ) ) ) holds <br/><font color="Olive" title="b3">D</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</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">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st ( for <font color="Olive" title="b3">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> iff  ex <font color="Olive" title="b4">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b5">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Olive" title="b1">M</font> st <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b5">C</font> ) ) ) holds <br/><font color="Olive" title="b2">D</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">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">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st ( for <font color="Olive" title="b2">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> iff  ex <font color="Olive" title="b3">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b4">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b4">C</font> ) ) ) holds <br/><font color="Olive" title="b1">D</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</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">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> iff  ex <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b3">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b3">C</font> ) ) ) implies <font color="Maroon" title="c4">D</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> iff  ex <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b3">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b3">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="c4">D</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:27"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
for <font color="Olive" title="b1">K</font> being  <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font> holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">K</font> be   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</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="c5">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font> implies  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:27_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D14" title="SUPINF_2:def.14">SUPINF_2:def 14</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:27_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c5">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font></span><br/>

<a NAME="E4:27_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font></span><br/>

<a NAME="E1:27_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:27_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1"><span class="lab"><font color="Green" title="E21">A4</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>
<a NAME="E5:27_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b3">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b3">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b2">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font> )</span><br/>

<a NAME="E1:27_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:27_1_2"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b2">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E18">A1</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>
<a NAME="E6:27_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E1:27_1_3"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b2">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:27_1_3"/>
<font color="Maroon" title="c6">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#D4" target="_self" title="MEASURE3:def.4">Def4</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>
<a NAME="E7:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A7</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c6">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#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>
 
;<br/>
<a NAME="E8:27_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E11:27_1"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K3" title="MEASURE3:func.3">MeasPart</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#T18" target="_self" title="MEASURE3:th.18">Th18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E13:27_1"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#T19" target="_self" title="MEASURE3:th.19">Th19</a></span>;<br/>
<a NAME="E14:27_1"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> )</span><br/>

<a NAME="E1:27_1_4"/>
<font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K3" title="MEASURE3:func.3">MeasPart</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:27_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:27_1_4"/>
( <font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#D5" target="_self" title="MEASURE3:def.5">Def5</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>
<a NAME="E15:27_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E1:27_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:27_1_5"/>
<font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n</font> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27_1"><span class="lab"><font color="Green" title="E27">A10</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">then </span><span class="kw">consider </span><font color="Maroon" title="c10">L</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E17:27_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#T20" target="_self" title="MEASURE3:th.20">Th20</a></span>;<br/>
<a NAME="E18:27_1"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b2">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c11">B</font> =  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span>;<br/>
<span class="kw">take </span>
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b1">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font> )</span><br/>

<a NAME="E1:27_1_6"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_1"><span class="kw">proof </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> ; <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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_6_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span>
 ; <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>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:27_1_6_1"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Z</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_6_1"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<font color="Maroon" title="c13">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:27_1_6_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:27_1_6_1"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E8:27_1_6_1"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
<font color="Maroon" title="c13">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_6_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">n</font> = <font color="Maroon" title="c14">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_1_6_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">P</font> = <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font>;<br/>
<a NAME="E10:27_1_6_1"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a></span>;<br/>
<a NAME="E11:27_1_6_1"/>
 ex <font color="Olive" title="b1">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">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" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:27_1_6_1_1"/>
( <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E3:27_1_6_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E8:27_1_6_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E10:27_1_6_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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="E12:27_1_6_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
<a NAME="E2:27_1_6"/>
 ex <font color="Olive" title="b1">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:27_1_6_2"/>
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font> holds <br/><font color="Olive" title="b1">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">A</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">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font> implies <font color="Maroon" title="c12">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_6_2_1"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</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="c12">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E2:27_1_6_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:27_1_6_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A20</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_6_2_1"><span class="lab"><font color="Green" title="E31">A19</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="E4:27_1_6_2_1"/>
 <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure2.html#D1" title="MEASURE2:def.1">MEASURE2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">A</font> = <font color="Maroon" title="c12">A</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_6_2_1"><span class="lab"><font color="Green" title="E31">A19</font></span></a></span>;<br/>
<a NAME="E6:27_1_6_2_1"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:27_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E3:27_1_6_2_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:27_1_6_2_1"/>
<font color="Maroon" title="c12">A</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</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>
<a NAME="E2:27_1_6_2"/><span class="kw">then </span>
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure2.html#T16" title="MEASURE2:th.16">MEASURE2:16</a></span>;<br/>
<a NAME="E3:27_1_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A21</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">C</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E32">A22</font></span>:  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p3">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_6"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">C</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E6:27_1_6_2"/><span class="lab"><font color="Green" title="E33">A23</font></span>: 
<font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_6_2_3"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font></span>)</span></span><br/>

<a NAME="E2:27_1_6_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:27_1_6_2_3"/><span class="lab"><font color="Green" title="E35">A25</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Z</font>
 <span class="kw">and </span><br/><a NAME="E5:27_1_6_2_3"/><span class="lab"><font color="Green" title="E36">A26</font></span>: 
<font color="Maroon" title="c15">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:27_1_6_2_3"/><span class="lab"><font color="Green" title="E37">A27</font></span>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E8:27_1_6_2_3"/><span class="lab"><font color="Green" title="E38">A28</font></span>: 
<font color="Maroon" title="c15">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E5:27_1_6_2_3"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">n</font> = <font color="Maroon" title="c16">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_1_6_2_3"><span class="lab"><font color="Green" title="E37">A27</font></span></a></span>;<br/>
<a NAME="E10:27_1_6_2_3"/><span class="lab"><font color="Green" title="E39">A29</font></span>: 
not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_6_2_3"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E11:27_1_6_2_3"/>
not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_2_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:27_1_6_2_3_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:27_1_6_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A30</font></span>: 
<font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</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/>
<span class="kw">assume </span><a NAME="E3:27_1_6_2_3_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E4:27_1_6_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_1_6_2_3"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="txt" href="#E2:27_1_6_2_3_1"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
<a NAME="E12:27_1_6_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A31</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_6_2_3"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="txt" href="#E8:27_1_6_2_3"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E13:27_1_6_2_3"/>
 ex <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_2_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">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" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font> )</span><br/>

<a NAME="E1:27_1_6_2_3_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:27_1_6_2_3_2"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E12:27_1_6_2_3"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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="E14:27_1_6_2_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
<a NAME="E7:27_1_6_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font></span>)</span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_6_2_4"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font></span>)</span>
 ; <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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:27_1_6_2_4"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Z</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_6_2_4"/><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<font color="Maroon" title="c15">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:27_1_6_2_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:27_1_6_2_4"/><span class="lab"><font color="Green" title="E36">A34</font></span>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E8:27_1_6_2_4"/><span class="lab"><font color="Green" title="E37">A35</font></span>: 
<font color="Maroon" title="c15">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_6_2_4"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">n</font> = <font color="Maroon" title="c16">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_1_6_2_4"><span class="lab"><font color="Green" title="E36">A34</font></span></a></span>;<br/>
<a NAME="E10:27_1_6_2_4"/>
<font color="Maroon" title="c17">n</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> <font color="Maroon" title="c10">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_1_6_2_4"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:27_1_6_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A36</font></span>: 
<font color="Maroon" title="c10">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</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="E12:27_1_6_2_4"/>
<font color="Maroon" title="c9">H</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:27_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:27_1_6_2_4"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_6_2_4"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="txt" href="#E8:27_1_6_2_4"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E11:27_1_6_2_4"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
<a NAME="E8:27_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c10">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1_6_2"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E9:27_1_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">C</font> is    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_6_2"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="measure3.html#D3" target="_self" title="MEASURE3:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">C</font> being    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font><span class="kw"> such that </span><br/><a NAME="E11:27_1_6_2"/><span class="lab"><font color="Green" title="E34">A37</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_6_2"><span class="lab"><font color="Green" title="E32">A22</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">C</font></span><br/>

<span class="kw">thus </span><a NAME="E12:27_1_6_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:27_1_6_2"><span class="lab"><font color="Green" title="E34">A37</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">hence </span><a NAME="E3:27_1_6"/>
(  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b1">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">G</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font> )
 ; <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="E19:27_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E18">A1</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>
<a NAME="E3:27"/>
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> holds <br/><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</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="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> implies <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_2"/><span class="lab"><font color="Green" title="E20">A38</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</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="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font></span><br/>

<a NAME="E2:27_2"/>
 ex <font color="Olive" title="b1">Q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b2">W</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c6">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:27_2_1"/><span class="lab"><font color="Green" title="E21">A39</font></span>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E3:27_2_1"/><span class="lab"><font color="Green" title="E22">A40</font></span>: 
 ex <font color="Olive" title="b1">C</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E1:27_2"><span class="lab"><font color="Green" title="E20">A38</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">P</font> = <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">C</font> being    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font><span class="kw"> such that </span><br/><a NAME="E5:27_2_1"/><span class="lab"><font color="Green" title="E23">A41</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_2_1"><span class="lab"><font color="Green" title="E22">A40</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">G</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:27_2_1"/><span class="lab"><font color="Green" title="E24">A42</font></span>: 
<font color="Maroon" title="c9">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E8:27_2_1"/><span class="lab"><font color="Green" title="E25">A43</font></span>: 
<font color="Maroon" title="c8">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G</font>
 <span class="kw">and </span><br/><a NAME="E9:27_2_1"/><span class="lab"><font color="Green" title="E26">A44</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#D3" target="_self" title="MEASURE3:def.3">Def3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">G</font>;<br/>
<a NAME="E10:27_2_1"/><span class="lab"><font color="Green" title="E27">A45</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_2_1"><span class="lab"><font color="Green" title="E23">A41</font></span></a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<a NAME="E11:27_2_1"/><span class="lab"><font color="Green" title="E28">A46</font></span>: 
 ex <font color="Olive" title="b1">W</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c11">W</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">C</font></span>)</span>;<br/>
<a NAME="E1:27_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">W</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">C</font></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">W</font> = <font color="Maroon" title="c12">W</font> as    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_2_1"><span class="lab"><font color="Green" title="E24">A42</font></span></a>, <a class="txt" href="#E9:27_2_1"><span class="lab"><font color="Green" title="E26">A44</font></span></a>, <a class="ref" href="measure3.html#D3" target="_self" title="MEASURE3:def.3">Def3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">W</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="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c13">W</font></span><br/>

<span class="kw">thus </span><a NAME="E4:27_2_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c13">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_2_1"><span class="lab"><font color="Green" title="E25">A43</font></span></a>, <a class="txt" href="#E10:27_2_1"><span class="lab"><font color="Green" title="E27">A45</font></span></a>, <a class="ref" href="xboole_1.html#T117" title="XBOOLE_1:th.117">XBOOLE_1:117</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">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b1">W</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">W</font> )</span><br/>

<a NAME="E12:27_2_1"/>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_2_1"><span class="lab"><font color="Green" title="E21">A39</font></span></a>, <a class="ref" href="measure1.html#D3" title="MEASURE1:def.3">MEASURE1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:27_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp;  ex <font color="Olive" title="b1">W</font> being   <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of <font color="Maroon" title="c3">M</font> st <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_2_1"><span class="lab"><font color="Green" title="E24">A42</font></span></a>, <a class="txt" href="#E11:27_2_1"><span class="lab"><font color="Green" title="E28">A46</font></span></a>, <a class="ref" href="measure1.html#T20" title="MEASURE1:th.20">MEASURE1:20</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="E3:27_2"/>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E18">A1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">D9</font> = <font color="Maroon" title="c4">D</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a>   <a href="measure1.html#V3" title="MEASURE1:attr.3">sigma-additive</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="measure1.html#D3" title="MEASURE1:def.3">MEASURE1:def 3</a>, <a class="ref" href="measure1.html#D9" title="MEASURE1:def.9">MEASURE1:def 9</a></span>;<br/>
<a NAME="E5:27"/>
<font color="Maroon" title="c5">D9</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:27"/>
<font color="Maroon" title="c4">D</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>
 ; <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>
