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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be  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> ; <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">L</font> being  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a><br/> for <font color="Olive" title="b2">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b3">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="b3">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  holds <br/>for <font color="Olive" title="b4">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b3">d</font> holds   <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Olive" title="b4">q</font>,<font color="Olive" title="b2">O</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</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">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b2">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  holds <br/>for <font color="Olive" title="b3">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b2">d</font> holds   <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Olive" title="b3">q</font>,<font color="Olive" title="b1">O</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">O</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  holds <br/>for <font color="Olive" title="b2">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b1">d</font> holds   <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c3">O</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">d</font> be   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</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="c4">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  implies for <font color="Olive" title="b1">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c4">d</font> holds   <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c3">O</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:67"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c4">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</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">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c4">d</font> holds   <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c3">O</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">q</font> be    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c4">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">O</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> </span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span> <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,$1 is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> ;<br/>
<a NAME="E2:67"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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">S<sub>1</sub></font>[<font color="Maroon" title="c6">O1</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_1"/>
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</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">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font>]</span><br/>

<a NAME="E2:67_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A3</font></span>: 
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" title="LATTICE5:func.10">Quadr</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T19" target="_self" title="LATTICE5:th.19">Th19</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">LATTICE5:def 6</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">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">x9</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">y9</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <span class="p1">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T25" target="_self" title="LATTICE5:th.25">Th25</a></span>;<br/>
<span class="lab"><font color="Green" title="E52">A4</font></span>:  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span> = 
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11" title="LATTICE5:func.11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<font color="Maroon" title="c2">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" title="LATTICE5:func.10">Quadr</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T30" target="_self" title="LATTICE5:th.30">Th30</a></span>
<br/>.= 
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" title="LATTICE5:func.10">Quadr</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D16" target="_self" title="LATTICE5:def.16">Def16</a></span>
;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> = 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K10" title="LATTICE5:func.10">Quadr</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c10">y9</font>,<font color="Maroon" title="c9">x9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a></span>
;<br/> <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="E3:67"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">O2</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">O1</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <span class="kw">-&gt; </span>  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,$1</span>)</span>,<font color="Maroon" title="c2">L</font> =  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,$1;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">O2</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O1</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">O2</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67_2"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
( <font color="Maroon" title="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">O2</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )
 <span class="kw">and </span><br/><a NAME="E2:67_2"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> holds <br/> <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">O1</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</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">S<sub>1</sub></font>[<font color="Maroon" title="c6">O2</font>]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">CS</font> =  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">Ls</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E4:67_2"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">Ls</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> holds <br/><font color="Maroon" title="c8">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">O1</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></span>
<a NAME="E5:67_2"/>
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">Ls</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_2"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="ref" href="lattice5.html#T31" target="_self" title="LATTICE5:th.31">Th31</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">Ls</font></span>)</span> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,$1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">Ts</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><span class="kw"> such that </span><br/><a NAME="E8:67_2"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">Ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> holds <br/><font color="Maroon" title="c10">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">O1</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></span>
<a NAME="E9:67_2"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">Ts</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_2"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="lattice5.html#T26" target="_self" title="LATTICE5:th.26">Th26</a></span>;<br/>
<a NAME="E10:67_2"/>
<font color="Maroon" title="c9">f</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">LATTICE5:def 6</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="c9">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:67_2_1"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">x1</font>
 <span class="kw">and </span><br/><a NAME="E3:67_2_1"/><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">Ts</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">o1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:67_2_1"/><span class="lab"><font color="Green" title="E59">A13</font></span>: 
<font color="Maroon" title="c14">o1</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> <font color="Maroon" title="c10">Ts</font>
 <span class="kw">and </span><br/><a NAME="E6:67_2_1"/><span class="lab"><font color="Green" title="E60">A14</font></span>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_2_1"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:67_2_1"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">y1</font>
 <span class="kw">and </span><br/><a NAME="E9:67_2_1"/><span class="lab"><font color="Green" title="E62">A16</font></span>: 
<font color="Maroon" title="c15">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">Ts</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:67_2_1"/><span class="lab"><font color="Green" title="E63">A17</font></span>: 
<font color="Maroon" title="c16">o2</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> <font color="Maroon" title="c10">Ts</font>
 <span class="kw">and </span><br/><a NAME="E12:67_2_1"/><span class="lab"><font color="Green" title="E64">A18</font></span>: 
<font color="Maroon" title="c15">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67_2_1"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">o1</font> = <font color="Maroon" title="c14">o1</font>, <font color="Maroon" title="c18">o2</font> = <font color="Maroon" title="c16">o2</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_2_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E11:67_2_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a></span>;<br/>
<a NAME="E14:67_2_1"/><span class="lab"><font color="Green" title="E65">A19</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c17">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E2:67_2_1"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E5:67_2_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E6:67_2_1"><span class="lab"><font color="Green" title="E60">A14</font></span></a></span>;<br/>
<a NAME="E15:67_2_1"/><span class="lab"><font color="Green" title="E66">A20</font></span>: 
<font color="Maroon" title="c8">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c17">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E5:67_2_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">h1</font> = <font color="Maroon" title="c8">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">o1</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c17">o1</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E17:67_2_1"/><span class="lab"><font color="Green" title="E67">A21</font></span>: 
<font color="Maroon" title="c19">h1</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">x</font>, <font color="Maroon" title="c21">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c17">o1</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">LATTICE5:def 6</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="c19">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c20">x</font></span><br/>

<a NAME="E1:67_2_1_1"/><span class="lab"><font color="Green" title="E68">A22</font></span>: 
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c17">o1</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_2"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E5:67_2_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c19">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c21">y</font> = 
<span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c17">o1</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c21">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E5:67_2_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c17">o1</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_2_1_1"><span class="lab"><font color="Green" title="E68">A22</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c19">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c20">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E5:67_2_1"><span class="lab"><font color="Green" title="E59">A13</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="E18:67_2_1"/><span class="lab"><font color="Green" title="E68">A23</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">h1</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"><span class="p2">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c17">o1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c17">o1</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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E19:67_2_1"/><span class="lab"><font color="Green" title="E69">A24</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c18">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E8:67_2_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E11:67_2_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E12:67_2_1"><span class="lab"><font color="Green" title="E64">A18</font></span></a></span>;<br/>
<a NAME="E20:67_2_1"/><span class="lab"><font color="Green" title="E70">A25</font></span>: 
<font color="Maroon" title="c8">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c18">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E11:67_2_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">h2</font> = <font color="Maroon" title="c8">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o2</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c18">o2</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E22:67_2_1"/><span class="lab"><font color="Green" title="E71">A26</font></span>: 
<font color="Maroon" title="c20">h2</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font>, <font color="Maroon" title="c22">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c18">o2</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">LATTICE5:def 6</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="c20">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c21">x</font></span><br/>

<a NAME="E1:67_2_1_2"/><span class="lab"><font color="Green" title="E72">A27</font></span>: 
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c18">o2</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_2"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E11:67_2_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c20">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">y</font> = 
<span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c18">o2</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">x</font>,<font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E11:67_2_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c18">o2</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c21">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_2_1_2"><span class="lab"><font color="Green" title="E72">A27</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c20">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c22">y</font>,<font color="Maroon" title="c21">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E11:67_2_1"><span class="lab"><font color="Green" title="E63">A17</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="E23:67_2_1"/><span class="lab"><font color="Green" title="E72">A28</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">h2</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"><span class="p2">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c18">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c18">o2</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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_2_1_3"/>
( <font color="Maroon" title="c17">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c18">o2</font> or <font color="Maroon" title="c18">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c17">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:67_2_1_3_1"/>
<font color="Maroon" title="c17">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c18">o2</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">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font></span><br/><div class="add"><a NAME="E2:67_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A29</font></span>: 
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c17">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c18">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T32" target="_self" title="LATTICE5:th.32">Th32</a></span>;<br/><a NAME="E3:67_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A30</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67_2_1"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="txt" href="#E19:67_2_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E23:67_2_1"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E4:67_2_1_3_1"/>
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c18">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">Ls</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E11:67_2_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E20:67_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:67_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A31</font></span>: 
<font color="Maroon" title="c20">h2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:67_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">x9</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c22">y9</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c18">o2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E8:67_2_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E11:67_2_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E12:67_2_1"><span class="lab"><font color="Green" title="E64">A18</font></span></a>, <a class="txt" href="#E14:67_2_1"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="txt" href="#E2:67_2_1_3_1"><span class="lab"><font color="Green" title="E73">A29</font></span></a></span>;<br/><a NAME="E7:67_2_1_3_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67_2_1"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="txt" href="#E19:67_2_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E23:67_2_1"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="txt" href="#E2:67_2_1_3_1"><span class="lab"><font color="Green" title="E73">A29</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">hence </span><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> = 
<font color="Maroon" title="c20">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">x9</font>,<font color="Maroon" title="c22">y9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_2_1_3_1"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>
<br/>.= 
<font color="Maroon" title="c20">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c22">y9</font>,<font color="Maroon" title="c21">x9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:67_2_1"><span class="lab"><font color="Green" title="E71">A26</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_2_1_3_1"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="txt" href="#E3:67_2_1_3_1"><span class="lab"><font color="Green" title="E74">A30</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_2_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:67_2_1_3_2"/>
<font color="Maroon" title="c18">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c17">o1</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">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font></span><br/><div class="add"><a NAME="E2:67_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A32</font></span>: 
 <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c18">o2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c17">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T32" target="_self" title="LATTICE5:th.32">Th32</a></span>;<br/><a NAME="E3:67_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A33</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67_2_1"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="txt" href="#E19:67_2_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E18:67_2_1"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E4:67_2_1_3_2"/>
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c17">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">Ls</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E5:67_2_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E15:67_2_1"><span class="lab"><font color="Green" title="E66">A20</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:67_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A34</font></span>: 
<font color="Maroon" title="c19">h1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:67_2_1"><span class="lab"><font color="Green" title="E66">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">x9</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c22">y9</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c17">o1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E2:67_2_1"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E5:67_2_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E6:67_2_1"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="txt" href="#E19:67_2_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E2:67_2_1_3_2"><span class="lab"><font color="Green" title="E73">A32</font></span></a></span>;<br/><a NAME="E7:67_2_1_3_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67_2_1"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="txt" href="#E19:67_2_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E18:67_2_1"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="txt" href="#E2:67_2_1_3_2"><span class="lab"><font color="Green" title="E73">A32</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">hence </span><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> = 
<font color="Maroon" title="c19">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">x9</font>,<font color="Maroon" title="c22">y9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_2_1_3_2"><span class="lab"><font color="Green" title="E75">A34</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>
<br/>.= 
<font color="Maroon" title="c19">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c22">y9</font>,<font color="Maroon" title="c21">x9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:67_2_1"><span class="lab"><font color="Green" title="E67">A21</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_2_1_3_2"><span class="lab"><font color="Green" title="E75">A34</font></span></a>, <a class="txt" href="#E3:67_2_1_3_2"><span class="lab"><font color="Green" title="E74">A33</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>
;<br/> <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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:67_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">O2</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_2"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="ref" href="lattice5.html#T31" target="_self" title="LATTICE5:th.31">Th31</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="E4:67"/><span class="lab"><font color="Green" title="E52">A35</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K9" title="LATTICE5:func.9">ConsecutiveSet</a> <font color="Maroon" title="c1">A</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">LATTICE5:def 6</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">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">x9</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y9</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T24" target="_self" title="LATTICE5:th.24">Th24</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> = 
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c8">x9</font>,<font color="Maroon" title="c9">y9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T29" target="_self" title="LATTICE5:th.29">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c9">y9</font>,<font color="Maroon" title="c8">x9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T29" target="_self" title="LATTICE5:th.29">Th29</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="E5:67"/>
for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E52">A35</font></span></a>, <a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E51">A5</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:67"/>
 <a href="lattice5.html#K13" title="LATTICE5:func.13">ConsecutiveDelta</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">O</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 ; <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>
