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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">K</font> being  <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">RK</font> being   <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">K</font> holds   <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Olive" title="b1">G</font>,<font color="Olive" title="b3">RK</font> is    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</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"> for <font color="Olive" title="b1">K</font> being  <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">RK</font> being   <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">K</font> holds   <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Olive" title="b2">RK</font> is    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</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"> for <font color="Olive" title="b1">RK</font> being   <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">K</font> holds   <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Olive" title="b1">RK</font> is    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">RK</font> be    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font> is    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">X</font></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">IT</font> =  <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">V1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">D</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> as  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>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">A</font> = the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">VV</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E2:84"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:84"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E4:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c7">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/>
<a NAME="E5:84"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> iff  ex <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> iff  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> ) )</span><br/>

<div><span class="lab"><font color="Green" title="E46">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c11">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E2:84_1_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:84_1_1"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:84_1_1"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> )
 <span class="kw">and </span><br/><a NAME="E6:84_1_1"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E2:84_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c15">x2</font> = <font color="Maroon" title="c13">x2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K7" title="BCIALG_6:func.7">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="#E5:84_1_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">v1</font> = <font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c17">v2</font> = <font color="Maroon" title="c15">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E9:84_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c17">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E3:84_1_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E6:84_1_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E10:84_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x1</font> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c15">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def12</a></span>;<br/><span class="kw">hence </span><a NAME="E11:84_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</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>
<a NAME="E2:84_1"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> implies  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:84_1_2"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</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">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">y1</font> = <font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c12">y2</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> 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="E3:84_1_2"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c7">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1_2"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c11">y1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c12">y2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c10">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T2" title="BCIALG_1:th.2">BCIALG_1:2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:84_1_2"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E3:84_1_2"><span class="lab"><font color="Green" title="E48">A8</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:84_1"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> iff  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1"><span class="lab"><font color="Green" title="E46">A2</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><span class="kw">end;</span></div>
<a NAME="E6:84"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">A</font> = the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <font color="Maroon" title="c1">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">B</font> = the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span>; <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="c10">A</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c14">xx</font> = <font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c15">yy</font> = <font color="Maroon" title="c13">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K7" title="BCIALG_6:func.7">Union</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font> ;<br/><a NAME="E2:84_2"/>
( <font color="Maroon" title="c12">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 <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> &amp; <font color="Maroon" title="c13">y</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 <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">vx</font> = <font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c17">vy</font> = <font color="Maroon" title="c13">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E4:84_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><a NAME="E5:84_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">A</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">vx</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c17">vy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E6:84_2"/>
<font color="Maroon" title="c10">A</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def12</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>
<a NAME="E9:84"/><span class="kw">then </span>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> <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="c1">X</font> &amp; <font color="Maroon" title="c10">A</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="bcialg_1.html#U1" title="BCIALG_1:sel.1">InternalDiff</a> of <span class="p1">(<span class="default"><a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:84"/>
 <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">HK</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font> is    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</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" title="BCIALG_1:def.10">BCIALG_1:def 10</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/>


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