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

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span>  implies  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1"/><span class="lab"><font color="Green" title="E1">A1</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> </span><br/>

<a NAME="E2:2_1"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">z</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"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span>  or <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1"/>
<font color="Maroon" title="c2">z</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"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
<font color="Maroon" title="c3">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">X1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E6:2_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c4">X1</font>
 <span class="kw">and </span><br/><a NAME="E7:2_1_1"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">X1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">z1</font> = <font color="Maroon" title="c2">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="#E6:2_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E10:2_1_1"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<font color="Maroon" title="c6">y1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">z1</font>
 <span class="kw">and </span><br/><a NAME="E11:2_1_1"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
<font color="Maroon" title="c6">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="#E6:2_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="waybel_0.html#D15" onmouseover="rs('waybel_0/D15')" onmouseout="rh()">WAYBEL_0:def 15</a></span>;<br/>
<a NAME="E12:2_1_1"/>
<font color="Maroon" title="c4">X1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E7:2_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/>
<a NAME="E13:2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:2_1_1"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="waybel_0.html#D15" onmouseover="rs('waybel_0/D15')" onmouseout="rh()">WAYBEL_0:def 15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:2_1"/>
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</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"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font></span>)</span> or <font color="Maroon" title="c2">z</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"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_2"/><span class="lab"><font color="Green" title="E3">A9</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">z</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"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">z1</font> = <font color="Maroon" title="c2">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E4:2_1_2"/><span class="lab"><font color="Green" title="E4">A10</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">z1</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_2"/><span class="lab"><font color="Green" title="E5">A11</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_2"><span class="lab"><font color="Green" title="E3">A9</font></span></a>, <a class="ref" href="waybel_0.html#D15" onmouseover="rs('waybel_0/D15')" onmouseout="rh()">WAYBEL_0:def 15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E7:2_1_2"/><span class="lab"><font color="Green" title="E6">A12</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_2"/><span class="lab"><font color="Green" title="E7">A13</font></span>: 
<font color="Maroon" title="c5">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_2"><span class="lab"><font color="Green" title="E5">A11</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">X1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E10:2_1_2"/><span class="lab"><font color="Green" title="E8">A14</font></span>: 
<font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X1</font>
 <span class="kw">and </span><br/><a NAME="E11:2_1_2"/><span class="lab"><font color="Green" title="E9">A15</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">X1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E8:2_1_2"><span class="lab"><font color="Green" title="E7">A13</font></span></a></span>;<br/>
<a NAME="E12:2_1_2"/><span class="lab"><font color="Green" title="E10">A16</font></span>: 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c6">X1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a></span>;<br/>
<a NAME="E13:2_1_2"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c6">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_2"><span class="lab"><font color="Green" title="E4">A10</font></span></a>, <a class="txt" href="#E7:2_1_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="txt" href="#E10:2_1_2"><span class="lab"><font color="Green" title="E8">A14</font></span></a>, <a class="ref" href="waybel_0.html#D15" onmouseover="rs('waybel_0/D15')" onmouseout="rh()">WAYBEL_0:def 15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:2_1_2"/>
<font color="Maroon" title="c2">z</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"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_2"><span class="lab"><font color="Green" title="E10">A16</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:2_1"/>
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">X</font>] </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
