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<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="firebrick">::  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>  st <font color="Olive" title="b1">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>  holds <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="Olive" title="b1">L</font>  st <font color="Olive" title="b2">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  &amp; <font color="Olive" title="b2">d</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  holds <br/> for <font color="Olive" title="b3">O</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><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="b2">d</font>  st <font color="Olive" title="b3">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Olive" title="b2">d</font> holds <br/> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Olive" title="b4">q</font>,<font color="Olive" title="b3">O</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>  implies  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>  &amp; <font color="Olive" title="b1">d</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  holds <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">q</font> being    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b1">d</font>  st <font color="Olive" title="b2">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Olive" title="b1">d</font> holds <br/> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Olive" title="b3">q</font>,<font color="Olive" title="b2">O</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c2">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>  &amp; <font color="Olive" title="b1">d</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  holds <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">q</font> being    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b1">d</font>  st <font color="Olive" title="b2">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Olive" title="b1">d</font> holds <br/> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Olive" title="b3">q</font>,<font color="Olive" title="b2">O</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  &amp; <font color="Maroon" title="c3">d</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  implies  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">q</font> being    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c3">d</font>  st <font color="Olive" title="b1">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font> holds <br/> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">O</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:41"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c3">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 <span class="kw">and </span><br/><a NAME="E3:41"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c3">d</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">q</font> being    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c3">d</font>  st <font color="Olive" title="b1">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font> holds <br/> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">O</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">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="firebrick">::  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="c3">d</font>  st <font color="Maroon" title="c4">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font> holds <br/> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">O</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</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="c3">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font> implies  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">O</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</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>( $1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font> implies  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,$1) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  );<br/>
<a NAME="E4:41"/><span class="lab"><font color="Green" title="E37">A4</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="41_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="firebrick">::  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><span class="kw">that </span><br/><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
( <font color="Maroon" title="c6">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font> implies  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  )
 <span class="kw">and </span><br/><a NAME="E2:41_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> </span><br/>

<a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon" title="c6">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<a NAME="E4:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c6">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="c5">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#T25" title="LATTICE5:th.25">LATTICE5:25</a></span>;<br/>
<a NAME="E5:41_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E6:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>, <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">a9</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">b9</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D13" title="LATTICE5:def.13">LATTICE5:def 13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="firebrick">:: according to </font></span><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font>) <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<span class="p3">(<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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<span class="p3">(<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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font>)</span>)</span></span><br/>

<a NAME="E7:41_1"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="lattice8.html#T26" target="_self" title="LATTICE8:th.26">Th27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x9</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c11">y9</font> = <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c12">z9</font> = <font color="Maroon" title="c9">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="lattice8.html#T15" target="_self" title="LATTICE8:th.15">Th16</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">f</font> =  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c14">X</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Y</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">a</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> , <font color="Maroon" title="c17">b</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E10:41_1"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c3">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#T23" target="_self" title="LATTICE8:th.23">Th24</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">u</font>, <font color="Maroon" title="c19">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c20">a9</font>, <font color="Maroon" title="c21">b9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E12:41_1"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c18">u</font>,<font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">a9</font>,<font color="Maroon" title="c21">b9</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span>
 <span class="kw">and </span><br/><a NAME="E13:41_1"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c18">u</font>,<font color="Maroon" title="c19">v</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">a9</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c21">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_1"><span class="lab"><font color="Green" title="E42">A9</font></span></a></span>;<br/>
<a NAME="E14:41_1"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
 <a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c18">u</font>,<font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">a9</font>,<font color="Maroon" title="c21">b9</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E12:41_1"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="ref" href="lattice8.html#D6" target="_self" title="LATTICE8:def.6">Def7</a></span>;<br/>
<a NAME="E15:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A15</font></span>: 
( <font color="Maroon" title="c18">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  &amp; <font color="Maroon" title="c19">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D8" title="MCART_1:def.8">MCART_1:def 8</a>, <a class="ref" href="mcart_1.html#D9" title="MCART_1:def.9">MCART_1:def 9</a></span>;<br/>
<a NAME="E16:41_1"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
( <font color="Maroon" title="c20">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> &amp; <font color="Maroon" title="c21">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:41_1"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="mcart_1.html#D10" title="MCART_1:def.10">MCART_1:def 10</a>, <a class="ref" href="mcart_1.html#D11" title="MCART_1:def.11">MCART_1:def 11</a></span>;<br/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c18">u</font>,<font color="Maroon" title="c19">v</font>) = 
<font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">u</font>,<font color="Maroon" title="c19">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:41_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E10:41_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
;<br/>
<a NAME="E18:41_1"/><span class="kw">then </span>
 <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E7:41_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E13:41_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E16:41_1"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="ref" href="lattice8.html#T12" target="_self" title="LATTICE8:th.12">Th12</a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E19:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A17</font></span>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c10">x9</font>,<font color="Maroon" title="c12">z9</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<span class="p3">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c10">x9</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<span class="p3">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>)</span>)</span>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">y9</font>,<font color="Maroon" title="c12">z9</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<span class="p1">(<span class="default"><a href="lattice5.html#K10" title="LATTICE5:func.10">BiFun</a> (<span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</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="lattice8.html#T19" target="_self" title="LATTICE8:th.19">Th20</a></span>
<br/>.= 
 <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</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#D15" title="LATTICE5:def.15">LATTICE5:def 15</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E21:41_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<span class="p3">(<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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<span class="p3">(<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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:41_1"><span class="lab"><font color="Green" title="E50">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:41"/><span class="lab"><font color="Green" title="E38">A18</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#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">O1</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; (  for <font color="Olive" title="b2">O2</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">O2</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,$1)</span>)</span>,<font color="Maroon" title="c2">L</font> =  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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="firebrick">::  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">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#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">O2</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="E1:41_2"/><span class="lab"><font color="Green" title="E39">A19</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="E2:41_2"/><span class="lab"><font color="Green" title="E40">A20</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> &amp; <font color="Olive" title="b1">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font> holds <br/> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">O1</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 <span class="kw">and </span><br/><a NAME="E3:41_2"/><span class="lab"><font color="Green" title="E41">A21</font></span>: 
<font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O2</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">CS</font> =  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="E5:41_2"/><span class="lab"><font color="Green" title="E42">A22</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="E6:41_2"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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="#E1:41_2"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="ref" href="lattice8.html#T20" target="_self" title="LATTICE8:th.20">Th21</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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="E9:41_2"/><span class="lab"><font color="Green" title="E43">A23</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="E10:41_2"/><span class="lab"><font color="Green" title="E44">A24</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="#E1:41_2"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="lattice8.html#T16" target="_self" title="LATTICE8:th.16">Th17</a></span>;<br/>
<a NAME="E11:41_2"/>
<font color="Maroon" title="c9">f</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_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>, <font color="Maroon" title="c13">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font>); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>) <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c14">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:41_2_1"/><span class="lab"><font color="Green" title="E45">A25</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">X</font>
 <span class="kw">and </span><br/><a NAME="E3:41_2_1"/><span class="lab"><font color="Green" title="E46">A26</font></span>: 
<font color="Maroon" title="c14">X</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="#E10:41_2"><span class="lab"><font color="Green" title="E44">A24</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="c15">o1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:41_2_1"/><span class="lab"><font color="Green" title="E47">A27</font></span>: 
<font color="Maroon" title="c15">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="E6:41_2_1"/><span class="lab"><font color="Green" title="E48">A28</font></span>: 
<font color="Maroon" title="c14">X</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="c15">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_2_1"><span class="lab"><font color="Green" title="E46">A26</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:41_2_1"/><span class="lab"><font color="Green" title="E49">A29</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font>
 <span class="kw">and </span><br/><a NAME="E9:41_2_1"/><span class="lab"><font color="Green" title="E50">A30</font></span>: 
<font color="Maroon" title="c16">Y</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="#E10:41_2"><span class="lab"><font color="Green" title="E44">A24</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="c17">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:41_2_1"/><span class="lab"><font color="Green" title="E51">A31</font></span>: 
<font color="Maroon" title="c17">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="E12:41_2_1"/><span class="lab"><font color="Green" title="E52">A32</font></span>: 
<font color="Maroon" title="c16">Y</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="c17">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2_1"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:41_2_1"/><span class="lab"><font color="Green" title="E53">A33</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Z</font>
 <span class="kw">and </span><br/><a NAME="E15:41_2_1"/><span class="lab"><font color="Green" title="E54">A34</font></span>: 
<font color="Maroon" title="c18">Z</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="#E10:41_2"><span class="lab"><font color="Green" title="E44">A24</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="c19">o3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:41_2_1"/><span class="lab"><font color="Green" title="E55">A35</font></span>: 
<font color="Maroon" title="c19">o3</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="E18:41_2_1"/><span class="lab"><font color="Green" title="E56">A36</font></span>: 
<font color="Maroon" title="c18">Z</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="c19">o3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:41_2_1"><span class="lab"><font color="Green" title="E54">A34</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">o1</font> = <font color="Maroon" title="c15">o1</font>, <font color="Maroon" title="c21">o2</font> = <font color="Maroon" title="c17">o2</font>, <font color="Maroon" title="c22">o3</font> = <font color="Maroon" title="c19">o3</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="#E5:41_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="txt" href="#E17:41_2_1"><span class="lab"><font color="Green" title="E55">A35</font></span></a></span>;<br/>
<a NAME="E20:41_2_1"/><span class="lab"><font color="Green" title="E57">A37</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E2:41_2_1"><span class="lab"><font color="Green" title="E45">A25</font></span></a>, <a class="txt" href="#E5:41_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="txt" href="#E6:41_2_1"><span class="lab"><font color="Green" title="E48">A28</font></span></a></span>;<br/>
<a NAME="E21:41_2_1"/><span class="lab"><font color="Green" title="E58">A38</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="c22">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c22">o3</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E17:41_2_1"><span class="lab"><font color="Green" title="E55">A35</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">h3</font> = <font color="Maroon" title="c8">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">o3</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>)</span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E23:41_2_1"/><span class="lab"><font color="Green" title="E59">A39</font></span>: 
<font color="Maroon" title="c23">h3</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font>, <font color="Maroon" title="c25">y</font>, <font color="Maroon" title="c26">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">h3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c26">z</font>) <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">h3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">h3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c25">y</font>,<font color="Maroon" title="c26">z</font>)</span>)</span></span><br/>

<a NAME="E1:41_2_1_1"/>
<font color="Maroon" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_2"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E17:41_2_1"><span class="lab"><font color="Green" title="E55">A35</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E2:41_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A40</font></span>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c22">o3</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E17:41_2_1"><span class="lab"><font color="Green" title="E55">A35</font></span></a></span>;<br/>
<a NAME="E3:41_2_1_1"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c22">o3</font>) <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a> <font color="Maroon" title="c23">h3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E17:41_2_1"><span class="lab"><font color="Green" title="E55">A35</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:41_2_1_1"/>
<font color="Maroon" title="c23">h3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c26">z</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">h3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">h3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c25">y</font>,<font color="Maroon" title="c26">z</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2_1_1"><span class="lab"><font color="Green" title="E60">A40</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:41_2_1"/><span class="lab"><font color="Green" title="E60">A41</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c23">h3</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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</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="E25:41_2_1"/><span class="lab"><font color="Green" title="E61">A42</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E14:41_2_1"><span class="lab"><font color="Green" title="E53">A33</font></span></a>, <a class="txt" href="#E17:41_2_1"><span class="lab"><font color="Green" title="E55">A35</font></span></a>, <a class="txt" href="#E18:41_2_1"><span class="lab"><font color="Green" title="E56">A36</font></span></a></span>;<br/>
<a NAME="E26:41_2_1"/><span class="lab"><font color="Green" title="E62">A43</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="c21">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">o2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">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="c21">o2</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>)</span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E28:41_2_1"/><span class="lab"><font color="Green" title="E63">A44</font></span>: 
<font color="Maroon" title="c24">h2</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</font>, <font color="Maroon" title="c26">y</font>, <font color="Maroon" title="c27">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c27">z</font>) <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">y</font>,<font color="Maroon" title="c27">z</font>)</span>)</span></span><br/>

<a NAME="E1:41_2_1_2"/>
<font color="Maroon" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_2"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E2:41_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A45</font></span>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">o2</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a></span>;<br/>
<a NAME="E3:41_2_1_2"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">o2</font>) <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a> <font color="Maroon" title="c24">h2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:41_2_1_2"/>
<font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c27">z</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">y</font>,<font color="Maroon" title="c27">z</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2_1_2"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E29:41_2_1"/><span class="lab"><font color="Green" title="E64">A46</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">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/>
<a NAME="E30:41_2_1"/><span class="lab"><font color="Green" title="E65">A47</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="c20">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c20">o1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E5:41_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">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="c20">o1</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>)</span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E32:41_2_1"/><span class="lab"><font color="Green" title="E66">A48</font></span>: 
<font color="Maroon" title="c25">h1</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">x</font>, <font color="Maroon" title="c27">y</font>, <font color="Maroon" title="c28">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">h1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c28">z</font>) <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c27">y</font>,<font color="Maroon" title="c28">z</font>)</span>)</span></span><br/>

<a NAME="E1:41_2_1_3"/>
<font color="Maroon" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_2"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E5:41_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E2:41_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A49</font></span>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c20">o1</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E5:41_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a></span>;<br/>
<a NAME="E3:41_2_1_3"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c20">o1</font>) <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a> <font color="Maroon" title="c25">h1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E5:41_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:41_2_1_3"/>
<font color="Maroon" title="c25">h1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c28">z</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c27">y</font>,<font color="Maroon" title="c28">z</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2_1_3"><span class="lab"><font color="Green" title="E67">A49</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E33:41_2_1"/><span class="lab"><font color="Green" title="E67">A50</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c25">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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">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="E34:41_2_1"/><span class="lab"><font color="Green" title="E68">A51</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E8:41_2_1"><span class="lab"><font color="Green" title="E49">A29</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="txt" href="#E12:41_2_1"><span class="lab"><font color="Green" title="E52">A32</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_4"><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="E1:41_2_1_4"/>
( <font color="Maroon" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c22">o3</font> or <font color="Maroon" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c20">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_4_1"><span class="kw">suppose </span></a><a NAME="E1:41_2_1_4_1"/><span class="lab"><font color="Green" title="E69">A52</font></span>: 
<font color="Maroon" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c22">o3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>) <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:41_2_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A53</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th22</a></span>;<br/><span class="kw">thus </span><a NAME="E3:41_2_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_4_1_1_1"><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="E1:41_2_1_4_1_1_1"/>
( <font color="Maroon" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c22">o3</font> or <font color="Maroon" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c21">o2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_2_1_4_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A54</font></span>: 
<font color="Maroon" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c22">o3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c26">z9</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E14:41_2_1"><span class="lab"><font color="Green" title="E53">A33</font></span></a>, <a class="txt" href="#E17:41_2_1"><span class="lab"><font color="Green" title="E55">A35</font></span></a>, <a class="txt" href="#E18:41_2_1"><span class="lab"><font color="Green" title="E56">A36</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">x9</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E2:41_2_1_4_1"><span class="lab"><font color="Green" title="E70">A53</font></span></a></span>;<br/><a NAME="E4:41_2_1_4_1_1_1_1"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c22">o3</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="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E17:41_2_1"><span class="lab"><font color="Green" title="E55">A35</font></span></a>, <a class="txt" href="#E21:41_2_1"><span class="lab"><font color="Green" title="E58">A38</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E5:41_2_1_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A55</font></span>: 
<font color="Maroon" title="c23">h3</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="#E21:41_2_1"><span class="lab"><font color="Green" title="E58">A38</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E6:41_2_1_4_1_1_1_1"/><span class="lab"><font color="Green" title="E73">A56</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E71">A54</font></span></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th22</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a></span>;<br/><a NAME="E8:41_2_1_4_1_1_1_1"/>
<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="c13">z</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="c23">h3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E24:41_2_1"><span class="lab"><font color="Green" title="E60">A41</font></span></a>, <a class="txt" href="#E6:41_2_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E73">A56</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E9:41_2_1_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A57</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c28">y9</font>,<font color="Maroon" title="c26">z9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E72">A55</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E10:41_2_1_4_1_1_1_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="c13">z</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="c23">h3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E24:41_2_1"><span class="lab"><font color="Green" title="E60">A41</font></span></a>, <a class="txt" href="#E2:41_2_1_4_1"><span class="lab"><font color="Green" title="E70">A53</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E11:41_2_1_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A58</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c27">x9</font>,<font color="Maroon" title="c26">z9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E72">A55</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E12:41_2_1_4_1_1_1_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="c23">h3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E24:41_2_1"><span class="lab"><font color="Green" title="E60">A41</font></span></a>, <a class="txt" href="#E2:41_2_1_4_1"><span class="lab"><font color="Green" title="E70">A53</font></span></a>, <a class="txt" href="#E6:41_2_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E73">A56</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E13:41_2_1_4_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c23">h3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c27">x9</font>,<font color="Maroon" title="c28">y9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E72">A55</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E14:41_2_1_4_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:41_2_1"><span class="lab"><font color="Green" title="E59">A39</font></span></a>, <a class="txt" href="#E9:41_2_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E74">A57</font></span></a>, <a class="txt" href="#E11:41_2_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E75">A58</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_2_1_4_1_1_1_2"/><span class="lab"><font color="Green" title="E71">A59</font></span>: 
<font color="Maroon" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c21">o2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c26">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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E8:41_2_1"><span class="lab"><font color="Green" title="E49">A29</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="txt" href="#E12:41_2_1"><span class="lab"><font color="Green" title="E52">A32</font></span></a></span>;<br/><a NAME="E3:41_2_1_4_1_1_1_2"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">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="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="txt" href="#E26:41_2_1"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:41_2_1_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A60</font></span>: 
<font color="Maroon" title="c24">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="#E26:41_2_1"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E5:41_2_1_4_1_1_1_2"/><span class="lab"><font color="Green" title="E73">A61</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E71">A59</font></span></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th22</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">z9</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a></span>;<br/><a NAME="E7:41_2_1_4_1_1_1_2"/>
<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="c13">z</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="c24">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E29:41_2_1"><span class="lab"><font color="Green" title="E64">A46</font></span></a>, <a class="txt" href="#E5:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E73">A61</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E8:41_2_1_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A62</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">y9</font>,<font color="Maroon" title="c27">z9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E72">A60</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E9:41_2_1_4_1_1_1_2"/>
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2_1_4_1"><span class="lab"><font color="Green" title="E69">A52</font></span></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th22</a></span>;<br/><a NAME="E10:41_2_1_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A63</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E73">A61</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">x9</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a></span>;<br/><a NAME="E12:41_2_1_4_1_1_1_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="c24">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E29:41_2_1"><span class="lab"><font color="Green" title="E64">A46</font></span></a>, <a class="txt" href="#E10:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E13:41_2_1_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A64</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c28">x9</font>,<font color="Maroon" title="c26">y9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E72">A60</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E14:41_2_1_4_1_1_1_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="c13">z</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="c24">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E29:41_2_1"><span class="lab"><font color="Green" title="E64">A46</font></span></a>, <a class="txt" href="#E5:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E73">A61</font></span></a>, <a class="txt" href="#E10:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E15:41_2_1_4_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c28">x9</font>,<font color="Maroon" title="c27">z9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E72">A60</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E16:41_2_1_4_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:41_2_1"><span class="lab"><font color="Green" title="E63">A44</font></span></a>, <a class="txt" href="#E13:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E76">A64</font></span></a>, <a class="txt" href="#E8:41_2_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E74">A62</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_4_2"><span class="kw">suppose </span></a><a NAME="E1:41_2_1_4_2"/><span class="lab"><font color="Green" title="E69">A65</font></span>: 
<font color="Maroon" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c20">o1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>) <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:41_2_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A66</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th22</a></span>;<br/><span class="kw">thus </span><a NAME="E3:41_2_1_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_4_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_4_2_1_1"><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="E1:41_2_1_4_2_1_1"/>
( <font color="Maroon" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c21">o2</font> or <font color="Maroon" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c20">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_4_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_2_1_4_2_1_1_1"/><span class="lab"><font color="Green" title="E71">A67</font></span>: 
<font color="Maroon" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c21">o2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c26">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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E8:41_2_1"><span class="lab"><font color="Green" title="E49">A29</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="txt" href="#E12:41_2_1"><span class="lab"><font color="Green" title="E52">A32</font></span></a></span>;<br/><a NAME="E3:41_2_1_4_2_1_1_1"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">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="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E11:41_2_1"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="txt" href="#E26:41_2_1"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:41_2_1_4_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A68</font></span>: 
<font color="Maroon" title="c24">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="#E26:41_2_1"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E5:41_2_1_4_2_1_1_1"/><span class="lab"><font color="Green" title="E73">A69</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E71">A67</font></span></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th22</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">x9</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a></span>;<br/><a NAME="E7:41_2_1_4_2_1_1_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="c24">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E29:41_2_1"><span class="lab"><font color="Green" title="E64">A46</font></span></a>, <a class="txt" href="#E5:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E73">A69</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E8:41_2_1_4_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A70</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c27">x9</font>,<font color="Maroon" title="c26">y9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E72">A68</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E9:41_2_1_4_2_1_1_1"/>
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2_1_4_2"><span class="lab"><font color="Green" title="E69">A65</font></span></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th22</a></span>;<br/><a NAME="E10:41_2_1_4_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A71</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E73">A69</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">z9</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a></span>;<br/><a NAME="E12:41_2_1_4_2_1_1_1"/>
<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="c13">z</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="c24">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E29:41_2_1"><span class="lab"><font color="Green" title="E64">A46</font></span></a>, <a class="txt" href="#E10:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E75">A71</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E13:41_2_1_4_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A72</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">y9</font>,<font color="Maroon" title="c28">z9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E72">A68</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E14:41_2_1_4_2_1_1_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="c13">z</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="c24">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E29:41_2_1"><span class="lab"><font color="Green" title="E64">A46</font></span></a>, <a class="txt" href="#E5:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E73">A69</font></span></a>, <a class="txt" href="#E10:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E75">A71</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E15:41_2_1_4_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c27">x9</font>,<font color="Maroon" title="c28">z9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E72">A68</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E16:41_2_1_4_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:41_2_1"><span class="lab"><font color="Green" title="E63">A44</font></span></a>, <a class="txt" href="#E8:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E74">A70</font></span></a>, <a class="txt" href="#E13:41_2_1_4_2_1_1_1"><span class="lab"><font color="Green" title="E76">A72</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1_4_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_2_1_4_2_1_1_2"/><span class="lab"><font color="Green" title="E71">A73</font></span>: 
<font color="Maroon" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c20">o1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c26">x9</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E2:41_2_1"><span class="lab"><font color="Green" title="E45">A25</font></span></a>, <a class="txt" href="#E5:41_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="txt" href="#E6:41_2_1"><span class="lab"><font color="Green" title="E48">A28</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">z9</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E2:41_2_1_4_2"><span class="lab"><font color="Green" title="E70">A66</font></span></a></span>;<br/><a NAME="E4:41_2_1_4_2_1_1_2"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c20">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="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E5:41_2_1"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="txt" href="#E30:41_2_1"><span class="lab"><font color="Green" title="E65">A47</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E5:41_2_1_4_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A74</font></span>: 
<font color="Maroon" title="c25">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="#E30:41_2_1"><span class="lab"><font color="Green" title="E65">A47</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/><a NAME="E6:41_2_1_4_2_1_1_2"/><span class="lab"><font color="Green" title="E73">A75</font></span>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2_1_4_2_1_1_2"><span class="lab"><font color="Green" title="E71">A73</font></span></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th22</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">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="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a></span>;<br/><a NAME="E8:41_2_1_4_2_1_1_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="c25">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E33:41_2_1"><span class="lab"><font color="Green" title="E67">A50</font></span></a>, <a class="txt" href="#E6:41_2_1_4_2_1_1_2"><span class="lab"><font color="Green" title="E73">A75</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E9:41_2_1_4_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A76</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c25">h1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">x9</font>,<font color="Maroon" title="c28">y9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1_4_2_1_1_2"><span class="lab"><font color="Green" title="E72">A74</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E10:41_2_1_4_2_1_1_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="c13">z</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="c25">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2_1"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E33:41_2_1"><span class="lab"><font color="Green" title="E67">A50</font></span></a>, <a class="txt" href="#E2:41_2_1_4_2"><span class="lab"><font color="Green" title="E70">A66</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E11:41_2_1_4_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A77</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c26">x9</font>,<font color="Maroon" title="c27">z9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1_4_2_1_1_2"><span class="lab"><font color="Green" title="E72">A74</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E12:41_2_1_4_2_1_1_2"/>
<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="c13">z</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="c25">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:41_2_1"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="txt" href="#E25:41_2_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E33:41_2_1"><span class="lab"><font color="Green" title="E67">A50</font></span></a>, <a class="txt" href="#E2:41_2_1_4_2"><span class="lab"><font color="Green" title="E70">A66</font></span></a>, <a class="txt" href="#E6:41_2_1_4_2_1_1_2"><span class="lab"><font color="Green" title="E73">A75</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/><a NAME="E13:41_2_1_4_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c28">y9</font>,<font color="Maroon" title="c27">z9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1_4_2_1_1_2"><span class="lab"><font color="Green" title="E72">A74</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E14:41_2_1_4_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">f</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:41_2_1"><span class="lab"><font color="Green" title="E66">A48</font></span></a>, <a class="txt" href="#E9:41_2_1_4_2_1_1_2"><span class="lab"><font color="Green" title="E74">A76</font></span></a>, <a class="txt" href="#E11:41_2_1_4_2_1_1_2"><span class="lab"><font color="Green" title="E75">A77</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></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="E12:41_2"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O2</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="ref" href="lattice8.html#T20" target="_self" title="LATTICE8:th.20">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:41"/><span class="lab"><font color="Green" title="E39">A78</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="41_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:41_3"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</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="firebrick">:: according to </font></span><a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font>) <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font>)</span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">x9</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c10">y9</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c11">z9</font> = <font color="Maroon" title="c8">z</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="lattice8.html#T14" target="_self" title="LATTICE8:th.14">Th15</a></span>;<br/>
<a NAME="E3:41_3"/>
(  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c9">x9</font>,<font color="Maroon" title="c11">z9</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c9">x9</font>,<font color="Maroon" title="c10">y9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c10">y9</font>,<font color="Maroon" title="c11">z9</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="lattice8.html#T18" target="_self" title="LATTICE8:th.18">Th19</a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:41_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</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#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:41"/>
 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="#E6:41"><span class="lab"><font color="Green" title="E39">A78</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E38">A18</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:41"/>
( <font color="Maroon" title="c4">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K7" title="LATTICE5:func.7">DistEsti</a> <font color="Maroon" title="c3">d</font> implies  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">O</font>) is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
