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<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="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">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>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E</font>,<font color="Olive" title="b1">x</font> &amp; $2 <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="c2">E</font>,<font color="Olive" title="b2">y</font> holds <br/>$3 <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="c2">E</font>,<span class="p1">(<span class="default"><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></span>)</span>;<br/>
<a NAME="E1:72_1_1"/><span class="lab"><font color="Green" title="E34">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="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</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="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</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" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font>; <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">V</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Olive" title="b1">V</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:72_1_1_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c5">x1</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">and </span><br/><a NAME="E3:72_1_1_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c3">W1</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="c2">E</font>,<font color="Maroon" title="c5">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:72_1_1_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c6">y1</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">and </span><br/><a NAME="E6:72_1_1_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c4">W2</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="c2">E</font>,<font color="Maroon" title="c6">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c6">y1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_1_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E5:72_1_1_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">C</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">y1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">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">S<sub>1</sub></font>[<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c9">C</font>]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c3">W1</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="c2">E</font>,<font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c4">W2</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="c2">E</font>,<font color="Maroon" title="c11">y</font> implies <font color="Maroon" title="c9">C</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="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">y</font></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:72_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c3">W1</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="c2">E</font>,<font color="Maroon" title="c10">x</font>
 <span class="kw">and </span><br/><a NAME="E2:72_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon" title="c4">W2</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="c2">E</font>,<font color="Maroon" title="c11">y</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="c9">C</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="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">y</font></span>)</span></span><br/><a NAME="E3:72_1_1_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c8">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72_1_1_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E2:72_1_1_1_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></span>;<br/><a NAME="E4:72_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c11">y</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="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T26" title="EQREL_1:th.26">EQREL_1:26</a></span>;<br/><a NAME="E5:72_1_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c7">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_1_1_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E1:72_1_1_1_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></span>;<br/><a NAME="E6:72_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c10">x</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="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T26" title="EQREL_1:th.26">EQREL_1:26</a></span>;<br/><a NAME="E7:72_1_1_1_1"/><span class="kw">then </span>
<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="c7">x1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">y</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="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_1_1_1_1"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="bcialg_2.html#D9" target="_self" title="BCIALG_2:def.9">Def9</a></span>;<br/><a NAME="E8:72_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">y1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T26" title="EQREL_1:th.26">EQREL_1:26</a></span>;<br/><span class="kw">hence </span><a NAME="E9:72_1_1_1_1"/>
<font color="Maroon" title="c9">C</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="c2">E</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</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="E10:72_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c9">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">thus </span><a NAME="E2:72_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="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</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="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E</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" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E1:72_1_1"><span class="lab"><font color="Green" title="E34">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>
