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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c2">V9</font>, <font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c4">C9</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="c1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V9</font> &amp; <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C9</font> implies  <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font> is   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">K</font> =  <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">L</font> =  <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font>;<br/>
<a NAME="E1:22"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substlat.html#D4" title="SUBSTLAT:def.4">SUBSTLAT:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:22"/>
( <font color="Maroon" title="c1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V9</font> &amp; <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C9</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font> is   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span><br/>

<a NAME="E3:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A2</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="heyting3.html#T12" target="_self" title="HEYTING3:th.12">Th12</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</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"> ( <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="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font></span>)</span> implies <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="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E12">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="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">a9</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b9</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E3:22_1"/>
( <font color="Maroon" title="c7">a</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="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font></span>)</span> &amp; <font color="Maroon" title="c8">b</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="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E12">A3</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="kw">reconsider </span><font color="Maroon" title="c11">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c12">b1</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E11">A2</font></span></a></span>;<br/><a NAME="E5:22_1"/>
<font color="Maroon" title="c9">a9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/><a NAME="E6:22_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">a9</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">b9</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">Th15</a></span>;<br/><a NAME="E7:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E8:22_1"/>
<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="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</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="E5:22"/><span class="kw">then </span>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">K</font> =  <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font> as    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">K</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">K</font> )</span><br/><span class="kw">assume </span><a NAME="E1:22_2"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">K</font></span><br/><a NAME="E2:22_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E4:22_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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 <font color="Maroon" title="c7">K</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">K</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="#E1:22_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">p9</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c13">q9</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E6:22_2"/>
<font color="Maroon" title="c10">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/><a NAME="E7:22_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">q</font> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">Th15</a></span>;<br/><a NAME="E8:22_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">p9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E9:22_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</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="E8:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A6</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">K</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">K</font> )</span><br/><span class="kw">assume </span><a NAME="E1:22_3"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">K</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E3:22_3"/>
( <font color="Maroon" title="c8">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="c7">K</font> &amp; <font color="Maroon" title="c9">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 <font color="Maroon" title="c7">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_3"><span class="lab"><font color="Green" title="E13">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="kw">reconsider </span><font color="Maroon" title="c12">p9</font> = <font color="Maroon" title="c10">p</font>, <font color="Maroon" title="c13">q9</font> = <font color="Maroon" title="c11">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E11">A2</font></span></a></span>;<br/><a NAME="E5:22_3"/>
<font color="Maroon" title="c10">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_3"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/><a NAME="E6:22_3"/><span class="kw">then </span>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">p</font> holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">q</font> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">Th15</a></span>;<br/><a NAME="E7:22_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">p9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">Th15</a></span>;<br/><a NAME="E8:22_3"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">p9</font>,<font color="Maroon" title="c13">q9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E9:22_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_3"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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="E10:22"/><span class="kw">then </span>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></span>;<br/>
<a NAME="E11:22"/><span class="kw">then </span>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:22"/>
 <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">C</font> is   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <font color="Maroon" title="c2">V9</font>,<font color="Maroon" title="c4">C9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</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>
