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

<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">P</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">B</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Olive" title="b3">P</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Olive" title="b2">B</font></span>)</span> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Olive" title="b3">P</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">P</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">B</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Olive" title="b2">P</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Olive" title="b1">B</font></span>)</span> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Olive" title="b2">P</font></span><br/><b>let </b><font color="Maroon" title="c4">B</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c4">B</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Olive" title="b1">P</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Olive" title="b1">P</font></span><br/><b>let </b><font color="Maroon" title="c5">P</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c5">P</font> implies <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c5">P</font> )</span><br/>







<b>assume </b><a NAME="E1:131"/>
<font color="Maroon" title="c4">B</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c5">P</font></span><br/>

<a NAME="E2:131"/><b>then </b><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c5">P</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> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></i>;<br/>
<a NAME="E3:131"/>
<font color="Maroon" title="c5">P</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"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or <font color="Maroon" title="c6">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"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:131_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">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"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:131_1"/><i><font color="Green" title="E43">A2</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="txt" href="#E2:131"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:131_1"/>
 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="c6">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> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c7">Y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E3:131_1"><i><font color="Green" title="E43">A2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">Z</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span>;<br/>
<a NAME="E2:131_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" target="_self" title="FUNCT_2:def.1">Def1</a></i>;<br/>
<a NAME="E3:131_1_1"/><b>then </b><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/>
<a NAME="E4:131_1_1"/><i><font color="Green" title="E45">A4</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E3:131_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="funct_2.html#D11" target="_self" title="FUNCT_2:def.11">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E5:131_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">Y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:131_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="#E3:131_1_1"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="#E4:131_1_1"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="funct_2.html#D10" target="_self" title="FUNCT_2:def.10">Def10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:131_1"/>
<font color="Maroon" title="c6">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"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:131"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K6" title="FUNCT_2:func.6">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K7" title="FUNCT_2:func.7">.:</a> <font color="Maroon" title="c4">B</font></span>)</span> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
