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<span class="kw">set </span><font color="Maroon" title="c5">Z2</font> = <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">a</font>)</span>)</span> where <font color="Olive" title="b1">a</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">a</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/>
<span class="kw">set </span><font color="Maroon" title="c6">Z1</font> =  <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="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Olive" title="b1">a</font>)</span>)</span> where <font color="Olive" title="b1">a</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">a</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/>
<div><span class="lab"><font color="Green" title="E39">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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"> <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="b2">a</font>)</span>)</span> where <font color="Olive" title="b2">a</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="b2">a</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>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c7">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"> <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">a</font>)</span>)</span> where <font color="Olive" title="b1">a</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">a</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>  implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_1_1"/>
<font color="Maroon" title="c7">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"> <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">a</font>)</span>)</span> where <font color="Olive" title="b1">a</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">a</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> 
 ; <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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_1_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_1_1"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c8">g</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="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Olive" title="b1">a</font>)</span>)</span> where <font color="Olive" title="b1">a</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">a</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> 
 <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/><a NAME="E5:73_1_1_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">a</font>) &amp;  <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">a</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_1_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:73_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="#E3:73_1_1_1"><span class="lab"><font color="Green" title="E40">A2</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><span class="kw">end;</span></div>
<a NAME="E2:73_1_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">G</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/>
<a NAME="E3:73_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A4</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"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</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">a</font>)</span>)</span> where <font color="Olive" title="b1">a</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">a</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="E4:73_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</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="ref" href="eqrel_1.html#T5" onmouseover="rs('eqrel_1/T5')" onmouseout="rh()">EQREL_1:5</a></span>;<br/>
<a NAME="E5:73_1_1"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</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"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</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/>
<span class="kw">hence </span><a NAME="E6:73_1_1"/>
 <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="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c4">RK</font>,<font color="Olive" title="b1">a</font>)</span>)</span> where <font color="Olive" title="b1">a</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">a</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>  is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:73_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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/>


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