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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>),   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>),   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font>;<br/>
<a NAME="E1:75_1_1"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)  ex <font color="Olive" title="b3">v</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">w1</font>,<font color="Olive" title="b2">w2</font>,<font color="Olive" title="b3">v</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">w1</font>, <font color="Maroon" title="c6">w2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</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">  ex <font color="Olive" title="b1">v</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">w1</font>,<font color="Maroon" title="c6">w2</font>,<font color="Olive" title="b1">v</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E2:75_1_1_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c5">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Maroon" title="c7">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_6.html#E70"><span class="lab"><font color="Green" title="E39">Lm5</font></span></a></span>;<br/>
<a NAME="E3:75_1_1_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c5">w1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">RK</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E5:75_1_1_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c6">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Maroon" title="c8">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="bcialg_6.html#E70"><span class="lab"><font color="Green" title="E39">Lm5</font></span></a></span>;<br/>
<a NAME="E6:75_1_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D10" onmouseover="rs('bcialg_1/D10')" onmouseout="rh()">BCIALG_1:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b1</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E8:75_1_1_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c10">b1</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RK</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" onmouseover="rs('eqrel_1/D3')" onmouseout="rh()">EQREL_1:def 3</a></span>;<br/>
<a NAME="E9:75_1_1_1"/><span class="kw">then </span>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">b</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RK</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T34" target="_self" onmouseover="rs('bcialg_6/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E10:75_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A5</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">b</font></span>)</span>) <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="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Olive" title="b1">c</font>)</span>)</span> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> :  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Olive" title="b1">c</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c4">RK</font></span>)</span> </span> } </span> 
 
;<br/>
<a NAME="E11:75_1_1_1"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">w2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">RK</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">w1</font> = <font color="Maroon" title="c5">w1</font>, <font color="Maroon" title="c12">w2</font> = <font color="Maroon" title="c6">w2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E13:75_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c10">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">w1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c12">w2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">RK</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E11:75_1_1_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="bcialg_2.html#D9" onmouseover="rs('bcialg_2/D9')" onmouseout="rh()">BCIALG_2:def 9</a></span>;<br/>
<a NAME="E14:75_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">w1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c12">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c10">b1</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<a NAME="E15:75_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">w1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c12">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">b</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#T34" target="_self" onmouseover="rs('bcialg_6/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">C</font> = <font color="Maroon" title="c11">w1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c12">w2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:75_1_1_1"><span class="lab"><font color="Green" title="E44">A5</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">take </span>
<font color="Maroon" title="c13">C</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">w1</font>,<font color="Maroon" title="c6">w2</font>,<font color="Maroon" title="c13">C</font>]</span><br/>

<span class="kw">thus </span><a NAME="E17:75_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">w1</font>,<font color="Maroon" title="c6">w2</font>,<font color="Maroon" title="c13">C</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E2:75_1_1"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>)</span>)</span> st <br/> for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">w1</font>,<font color="Olive" title="b3">w2</font>,<font color="Olive" title="b1">B</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">w1</font>,<font color="Olive" title="b3">w2</font>)]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" onmouseover="rs('binop_1/S3')" onmouseout="rh()">BINOP_1:sch 3</a>(<a class="txt" href="#E1:75_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></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>
