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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being   <a href="lattice5.html#M2" title="LATTICE5:mode.2">ExtensionSeq</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>, <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">FS</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">FD</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b2">FS</font>,<font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b4">f</font> being  <a href="lattice5.html#NM1" title="LATTICE5:NM.1">Homomorphism</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Olive" title="b2">FS</font></span>)</span><br/> for <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">FS</font><br/> for <font color="Olive" title="b7">e1</font>, <font color="Olive" title="b8">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Olive" title="b2">FS</font><br/> for <font color="Olive" title="b9">x</font>, <font color="Olive" title="b10">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Olive" title="b3">FD</font> &amp; <font color="Olive" title="b2">FS</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b11">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Olive" title="b3">FD</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b11">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Olive" title="b7">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Olive" title="b4">f</font></span>)</span> &amp; <font color="Olive" title="b8">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Olive" title="b4">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b9">x</font>,<font color="Olive" title="b10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b8">e2</font> holds <br/> ex <font color="Olive" title="b11">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b2">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b11">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b9">x</font>,<font color="Olive" title="b10">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b11">F</font>,<font color="Olive" title="b7">e1</font>,<font color="Olive" title="b8">e2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be    <a href="lattice5.html#M2" title="LATTICE5:mode.2">ExtensionSeq</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>, <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">FS</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">FD</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Olive" title="b1">FS</font>,<font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="lattice5.html#NM1" title="LATTICE5:NM.1">Homomorphism</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Olive" title="b1">FS</font></span>)</span><br/> for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">FS</font><br/> for <font color="Olive" title="b6">e1</font>, <font color="Olive" title="b7">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Olive" title="b1">FS</font><br/> for <font color="Olive" title="b8">x</font>, <font color="Olive" title="b9">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Olive" title="b2">FD</font> &amp; <font color="Olive" title="b1">FS</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b10">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Olive" title="b2">FD</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b10">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Olive" title="b6">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Olive" title="b3">f</font></span>)</span> &amp; <font color="Olive" title="b7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Olive" title="b3">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b7">e2</font> holds <br/> ex <font color="Olive" title="b10">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b10">F</font>,<font color="Olive" title="b6">e1</font>,<font color="Olive" title="b7">e2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">FS</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">FD</font> being  <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c3">FS</font>,<font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="lattice5.html#NM1" title="LATTICE5:NM.1">Homomorphism</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c3">FS</font></span>)</span><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font><br/> for <font color="Olive" title="b5">e1</font>, <font color="Olive" title="b6">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><br/> for <font color="Olive" title="b7">x</font>, <font color="Olive" title="b8">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Olive" title="b1">FD</font> &amp; <font color="Maroon" title="c3">FS</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b9">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Olive" title="b1">FD</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b9">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Olive" title="b5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Olive" title="b2">f</font></span>)</span> &amp; <font color="Olive" title="b6">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Olive" title="b2">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b7">x</font>,<font color="Olive" title="b8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b6">e2</font> holds <br/> ex <font color="Olive" title="b9">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b7">x</font>,<font color="Olive" title="b8">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b9">F</font>,<font color="Olive" title="b5">e1</font>,<font color="Olive" title="b6">e2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">FD</font> be   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c3">FS</font>,<font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="lattice5.html#NM1" title="LATTICE5:NM.1">Homomorphism</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c3">FS</font></span>)</span><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font><br/> for <font color="Olive" title="b4">e1</font>, <font color="Olive" title="b5">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><br/> for <font color="Olive" title="b6">x</font>, <font color="Olive" title="b7">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c4">FD</font> &amp; <font color="Maroon" title="c3">FS</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b8">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c4">FD</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b8">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Olive" title="b4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Olive" title="b5">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">x</font>,<font color="Olive" title="b7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b5">e2</font> holds <br/> ex <font color="Olive" title="b8">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b6">x</font>,<font color="Olive" title="b7">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b8">F</font>,<font color="Olive" title="b4">e1</font>,<font color="Olive" title="b5">e2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="lattice5.html#NM1" title="LATTICE5:NM.1">Homomorphism</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c3">FS</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font><br/> for <font color="Olive" title="b3">e1</font>, <font color="Olive" title="b4">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><br/> for <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c4">FD</font> &amp; <font color="Maroon" title="c3">FS</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b7">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c4">FD</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b7">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Olive" title="b3">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Olive" title="b4">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">x</font>,<font color="Olive" title="b6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b4">e2</font> holds <br/> ex <font color="Olive" title="b7">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b5">x</font>,<font color="Olive" title="b6">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b7">F</font>,<font color="Olive" title="b3">e1</font>,<font color="Olive" title="b4">e2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c4">FD</font> &amp; <font color="Maroon" title="c3">FS</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b5">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c4">FD</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b5">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">e2</font> holds <br/> ex <font color="Olive" title="b5">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">e1</font>, <font color="Maroon" title="c9">e2</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c4">FD</font> &amp; <font color="Maroon" title="c3">FS</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b3">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c4">FD</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b3">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c9">e2</font> holds <br/> ex <font color="Olive" title="b3">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b3">F</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c4">FD</font> &amp; <font color="Maroon" title="c3">FS</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c4">FD</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c9">e2</font> implies  ex <font color="Olive" title="b1">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font> ) )</span><br/>

















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:85"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c4">FD</font>
 <span class="kw">and </span><br/><a NAME="E2:85"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
( <font color="Maroon" title="c3">FS</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c4">FD</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"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  )
 <span class="kw">and </span><br/><a NAME="E3:85"/><span class="lab"><font color="Green" title="E65">A3</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:85"/><span class="lab"><font color="Green" title="E66">A4</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:85"/><span class="lab"><font color="Green" title="E67">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c9">e2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font> )</span><br/>

<a NAME="E6:85"/><span class="lab"><font color="Green" title="E68">A6</font></span>: 
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5
 
;<br/>
<a NAME="E7:85"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c9">e2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">FS</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">v</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:85"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></span>;<br/>
<a NAME="E9:85"/><span class="lab"><font color="Green" title="E69">A7</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5
 
;<br/>
<a NAME="E10:85"/>
 <a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D2" title="YELLOW_2:def.2">YELLOW_2:def 2</a></span>;<br/>
<a NAME="E11:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A8</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:85"/><span class="lab"><font color="Green" title="E71">A9</font></span>: 
<font color="Maroon" title="c14">a</font> <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="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E14:85"/><span class="lab"><font color="Green" title="E72">A10</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:85"/><span class="lab"><font color="Green" title="E73">A11</font></span>: 
<font color="Maroon" title="c15">b</font> <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="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E17:85"/><span class="lab"><font color="Green" title="E74">A12</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E11:85"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c15">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:85"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="txt" href="#E16:85"><span class="lab"><font color="Green" title="E73">A11</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">a</font> = <font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c19">b</font> = <font color="Maroon" title="c17">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">e</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><span class="kw"> such that </span><br/><a NAME="E21:85"/><span class="lab"><font color="Green" title="E75">A13</font></span>: 
<font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c19">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E22:85"/><span class="lab"><font color="Green" title="E76">A14</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</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> <font color="Maroon" title="c20">e</font> iff <font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</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="Maroon" title="c18">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c19">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="lattice5.html#D9" target="_self" title="LATTICE5:def.9">Def9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">e19</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><span class="kw"> such that </span><br/><a NAME="E24:85"/><span class="lab"><font color="Green" title="E77">A15</font></span>: 
<font color="Maroon" title="c21">e19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a</font>
 <span class="kw">and </span><br/><a NAME="E25:85"/><span class="lab"><font color="Green" title="E78">A16</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</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> <font color="Maroon" title="c21">e19</font> iff <font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</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="Maroon" title="c18">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="lattice5.html#D9" target="_self" title="LATTICE5:def.9">Def9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">e29</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><span class="kw"> such that </span><br/><a NAME="E27:85"/><span class="lab"><font color="Green" title="E79">A17</font></span>: 
<font color="Maroon" title="c22">e29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b</font>
 <span class="kw">and </span><br/><a NAME="E28:85"/><span class="lab"><font color="Green" title="E80">A18</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</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> <font color="Maroon" title="c22">e29</font> iff <font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</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="Maroon" title="c19">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="lattice5.html#D9" target="_self" title="LATTICE5:def.9">Def9</a></span>;<br/>
<a NAME="E29:85"/><span class="lab"><font color="Green" title="E81">A19</font></span>: 
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5
 
;<br/>
<font color="Maroon" title="c20">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:85"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="ref" href="waybel_6.html#T2" title="WAYBEL_6:th.2">WAYBEL_6:2</a></span>
<br/>.= 
<font color="Maroon" title="c8">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:85"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E17:85"><span class="lab"><font color="Green" title="E74">A12</font></span></a>, <a class="ref" href="lattice5.html#T10" target="_self" title="LATTICE5:th.10">Th10</a></span>
;<br/>
<a NAME="E31:85"/><span class="kw">then </span>
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c13">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c19">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:85"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="txt" href="#E22:85"><span class="lab"><font color="Green" title="E76">A14</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">z1</font>, <font color="Maroon" title="c24">z2</font>, <font color="Maroon" title="c25">z3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font><span class="kw"> such that </span><br/><a NAME="E33:85"/><span class="lab"><font color="Green" title="E82">A20</font></span>: 
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a</font>
 <span class="kw">and </span><br/><a NAME="E34:85"/><span class="lab"><font color="Green" title="E83">A21</font></span>: 
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c24">z2</font>,<font color="Maroon" title="c25">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a</font>
 <span class="kw">and </span><br/><a NAME="E35:85"/><span class="lab"><font color="Green" title="E84">A22</font></span>: 
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c23">z1</font>,<font color="Maroon" title="c24">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font>
 <span class="kw">and </span><br/><a NAME="E36:85"/><span class="lab"><font color="Green" title="E85">A23</font></span>: 
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c25">z3</font>,<font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="lattice5.html#T45" target="_self" title="LATTICE5:th.45">Th45</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">z1</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">z2</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">z3</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> ) );<br/>
<a NAME="E37:85"/><span class="lab"><font color="Green" title="E86">A24</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 holds <br/> ex <font color="Olive" title="b2">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">w</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 implies  ex <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Olive" title="b1">w</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:85_2"/><span class="lab"><font color="Green" title="E87">A25</font></span>: 
<font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Olive" title="b1">w</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_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="E1:85_2_1"/>
( <font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_2"><span class="lab"><font color="Green" title="E87">A25</font></span></a>, <a class="txt" href="lattice5.html#E68"><span class="lab"><font color="Green" title="E58">Lm3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:85_2_1_1"/><span class="lab"><font color="Green" title="E88">A26</font></span>: 
<font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Olive" title="b1">w</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c27">w</font> = <font color="Maroon" title="c10">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]</span><br/><span class="kw">thus </span><a NAME="E2:85_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_2_1_1"><span class="lab"><font color="Green" title="E88">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:85_2_1_2"/><span class="lab"><font color="Green" title="E88">A27</font></span>: 
<font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Olive" title="b1">w</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c27">w</font> = <font color="Maroon" title="c23">z1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]</span><br/><span class="kw">thus </span><a NAME="E2:85_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_2_1_2"><span class="lab"><font color="Green" title="E88">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:85_2_1_3"/><span class="lab"><font color="Green" title="E88">A28</font></span>: 
<font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Olive" title="b1">w</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c27">w</font> = <font color="Maroon" title="c24">z2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]</span><br/><span class="kw">thus </span><a NAME="E2:85_2_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_2_1_3"><span class="lab"><font color="Green" title="E88">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:85_2_1_4"/><span class="lab"><font color="Green" title="E88">A29</font></span>: 
<font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Olive" title="b1">w</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c27">w</font> = <font color="Maroon" title="c25">z3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]</span><br/><span class="kw">thus </span><a NAME="E2:85_2_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_2_1_4"><span class="lab"><font color="Green" title="E88">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_1_5"><span class="kw">suppose </span></a><a NAME="E1:85_2_1_5"/><span class="lab"><font color="Green" title="E88">A30</font></span>: 
<font color="Maroon" title="c26">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Olive" title="b1">w</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c27">w</font> = <font color="Maroon" title="c11">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]</span><br/><span class="kw">thus </span><a NAME="E2:85_2_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_2_1_5"><span class="lab"><font color="Green" title="E88">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E38:85"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>,<font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font>] ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="#E37:85"><span class="lab"><font color="Green" title="E86">A24</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">h</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E40:85"/><span class="lab"><font color="Green" title="E87">A31</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5
 <span class="kw">and </span><br/><a NAME="E41:85"/><span class="lab"><font color="Green" title="E88">A32</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5 holds <br/>( ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">z1</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">z2</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 implies <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">z3</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 implies <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> ) )
 ;<br/>
<a NAME="E42:85"/><span class="lab"><font color="Green" title="E89">A33</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:85"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E43:85"/><span class="lab"><font color="Green" title="E90">A34</font></span>: 
5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5
 
;<br/>
<a NAME="E44:85"/><span class="lab"><font color="Green" title="E91">A35</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 5
 
;<br/>
<a NAME="E45:85"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">FS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c27">w</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="c26">h</font> or <font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:85_3"/>
<font color="Maroon" title="c27">w</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="c26">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:85_3"/><span class="lab"><font color="Green" title="E92">A36</font></span>: 
<font color="Maroon" title="c28">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">h</font>
 <span class="kw">and </span><br/><a NAME="E4:85_3"/><span class="lab"><font color="Green" title="E93">A37</font></span>: 
<font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_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="E1:85_3_1"/>
( <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:85"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="txt" href="#E3:85_3"><span class="lab"><font color="Green" title="E92">A36</font></span></a>, <a class="txt" href="lattice5.html#E68"><span class="lab"><font color="Green" title="E58">Lm3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:85_3_1_1"/>
<font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font></span><br/><div class="add"><a NAME="E2:85_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E9:85"><span class="lab"><font color="Green" title="E69">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:85_3_1_1"/>
<font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85_3"><span class="lab"><font color="Green" title="E93">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:85_3_1_2"/>
<font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font></span><br/><div class="add"><a NAME="E2:85_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">z1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E44:85"><span class="lab"><font color="Green" title="E91">A35</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:85_3_1_2"/>
<font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85_3"><span class="lab"><font color="Green" title="E93">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:85_3_1_3"/>
<font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font></span><br/><div class="add"><a NAME="E2:85_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E29:85"><span class="lab"><font color="Green" title="E81">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:85_3_1_3"/>
<font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85_3"><span class="lab"><font color="Green" title="E93">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_4"><span class="kw">suppose </span></a><a NAME="E1:85_3_1_4"/>
<font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font></span><br/><div class="add"><a NAME="E2:85_3_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">z3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E68">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:85_3_1_4"/>
<font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85_3"><span class="lab"><font color="Green" title="E93">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1_5"><span class="kw">suppose </span></a><a NAME="E1:85_3_1_5"/>
<font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font></span><br/><div class="add"><a NAME="E2:85_3_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E43:85"><span class="lab"><font color="Green" title="E90">A34</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:85_3_1_5"/>
<font color="Maroon" title="c27">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:85_3"><span class="lab"><font color="Green" title="E93">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">h</font> = <font color="Maroon" title="c26">h</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">h</font> = <font color="Maroon" title="c27">h</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:85"><span class="lab"><font color="Green" title="E87">A31</font></span></a></span>;<br/>
<a NAME="E48:85"/><span class="lab"><font color="Green" title="E92">A38</font></span>: 
<font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E9:85"><span class="lab"><font color="Green" title="E69">A7</font></span></a></span>;<br/>
<a NAME="E49:85"/><span class="lab"><font color="Green" title="E93">A39</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">h</font> holds <br/>( ( not <font color="Olive" title="b1">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Olive" title="b1">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">j</font> &amp; <font color="Maroon" title="c29">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">h</font> implies ( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:85_4"/><span class="lab"><font color="Green" title="E94">A40</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">j</font> &amp; <font color="Maroon" title="c29">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">h</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_4_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="E1:85_4_1"/>
( <font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:85"><span class="lab"><font color="Green" title="E89">A33</font></span></a>, <a class="txt" href="#E1:85_4"><span class="lab"><font color="Green" title="E94">A40</font></span></a>, <a class="txt" href="lattice5.html#E67"><span class="lab"><font color="Green" title="E57">Lm2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:85_4_1_1"/><span class="lab"><font color="Green" title="E95">A41</font></span>: 
<font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )</span><br/><div class="add"><a NAME="E2:85_4_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">u</font>,<font color="Maroon" title="c23">z1</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> <font color="Maroon" title="c21">e19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:85"><span class="lab"><font color="Green" title="E78">A16</font></span></a>, <a class="txt" href="#E33:85"><span class="lab"><font color="Green" title="E82">A20</font></span></a></span>;<br/><a NAME="E3:85_4_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c23">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">e19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E9:85"><span class="lab"><font color="Green" title="E69">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:85_4_1_1"/>
( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:85"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E24:85"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E44:85"><span class="lab"><font color="Green" title="E91">A35</font></span></a>, <a class="txt" href="#E1:85_4_1_1"><span class="lab"><font color="Green" title="E95">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:85_4_1_2"/><span class="lab"><font color="Green" title="E95">A42</font></span>: 
<font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )</span><br/><div class="add"><a NAME="E2:85_4_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">z2</font>,<font color="Maroon" title="c25">z3</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> <font color="Maroon" title="c21">e19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:85"><span class="lab"><font color="Green" title="E78">A16</font></span></a>, <a class="txt" href="#E34:85"><span class="lab"><font color="Green" title="E83">A21</font></span></a></span>;<br/><a NAME="E3:85_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A43</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c25">z3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">e19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E29:85"><span class="lab"><font color="Green" title="E81">A19</font></span></a></span>;<br/><a NAME="E4:85_4_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_4_1_2"><span class="lab"><font color="Green" title="E95">A42</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:85_4_1_2"/>
( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:85"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E24:85"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="txt" href="#E3:85_4_1_2"><span class="lab"><font color="Green" title="E96">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:85_4_1_3"/><span class="lab"><font color="Green" title="E95">A44</font></span>: 
<font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )</span><br/><div class="add"><a NAME="E2:85_4_1_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c23">z1</font>,<font color="Maroon" title="c24">z2</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> <font color="Maroon" title="c22">e29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:85"><span class="lab"><font color="Green" title="E80">A18</font></span></a>, <a class="txt" href="#E35:85"><span class="lab"><font color="Green" title="E84">A22</font></span></a></span>;<br/><a NAME="E3:85_4_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A45</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c24">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">e29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E44:85"><span class="lab"><font color="Green" title="E91">A35</font></span></a></span>;<br/><a NAME="E4:85_4_1_3"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_4_1_3"><span class="lab"><font color="Green" title="E95">A44</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:85_4_1_3"/>
( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:85"><span class="lab"><font color="Green" title="E74">A12</font></span></a>, <a class="txt" href="#E27:85"><span class="lab"><font color="Green" title="E79">A17</font></span></a>, <a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E29:85"><span class="lab"><font color="Green" title="E81">A19</font></span></a>, <a class="txt" href="#E3:85_4_1_3"><span class="lab"><font color="Green" title="E96">A45</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_4_1_4"><span class="kw">suppose </span></a><a NAME="E1:85_4_1_4"/><span class="lab"><font color="Green" title="E95">A46</font></span>: 
<font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )</span><br/><div class="add"><a NAME="E2:85_4_1_4"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">z3</font>,<font color="Maroon" title="c13">v</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> <font color="Maroon" title="c22">e29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:85"><span class="lab"><font color="Green" title="E80">A18</font></span></a>, <a class="txt" href="#E36:85"><span class="lab"><font color="Green" title="E85">A23</font></span></a></span>;<br/><a NAME="E3:85_4_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A47</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c13">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">e29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E68">A6</font></span></a></span>;<br/><a NAME="E4:85_4_1_4"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_4_1_4"><span class="lab"><font color="Green" title="E95">A46</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:85_4_1_4"/>
( ( not <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">e1</font> ) &amp; ( <font color="Maroon" title="c29">j</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:85"><span class="lab"><font color="Green" title="E74">A12</font></span></a>, <a class="txt" href="#E27:85"><span class="lab"><font color="Green" title="E79">A17</font></span></a>, <a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E43:85"><span class="lab"><font color="Green" title="E90">A34</font></span></a>, <a class="txt" href="#E3:85_4_1_4"><span class="lab"><font color="Green" title="E96">A47</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">take </span>
<font color="Maroon" title="c28">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E50:85"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:85"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font></span><br/>

<font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">h</font></span>)</span> = 
<font color="Maroon" title="c28">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:85"><span class="lab"><font color="Green" title="E87">A31</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c11">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:85"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="txt" href="#E43:85"><span class="lab"><font color="Green" title="E90">A34</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E52:85"/>
<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c8">e1</font>,<font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:85"><span class="lab"><font color="Green" title="E92">A38</font></span></a>, <a class="txt" href="#E49:85"><span class="lab"><font color="Green" title="E93">A39</font></span></a>, <a class="ref" href="lattice5.html#D3" target="_self" title="LATTICE5:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
