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<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="firebrick">::  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#K16" title="LATTICE5:func.16">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>   st <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="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>  holds <br/> <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="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>  is   <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></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#K16" title="LATTICE5:func.16">BasicDF</a> <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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>   st <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="b2">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> where <font color="Olive" title="b2">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>  holds <br/> <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="b2">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> where <font color="Olive" title="b2">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>  is   <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></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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  implies  <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>  is   <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> )</span><br/>





<span class="kw">assume </span><a NAME="E1:80"/><span class="lab"><font color="Green" title="E61">A1</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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>  is   <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></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">FS</font> = <font color="Maroon" title="c3">FS</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">A</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">FD</font> =  <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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>  implies <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:80_1"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="E2:80_1"/><span class="lab"><font color="Green" title="E63">A3</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c9">k</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> <span class="kw"> such that </span><br/><a NAME="E4:80_1"/><span class="lab"><font color="Green" title="E64">A4</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1"><span class="lab"><font color="Green" title="E62">A2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">l</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> <span class="kw"> such that </span><br/><a NAME="E6:80_1"/><span class="lab"><font color="Green" title="E65">A5</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1"><span class="lab"><font color="Green" title="E63">A3</font></span></a></span>;<br/><a NAME="E7:80_1"/>
( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">l</font> or <font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> )
 ;<br/><a NAME="E8:80_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c8">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="txt" href="#E6:80_1"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="lattice5.html#T39" target="_self" title="LATTICE5:th.39">Th42</a></span>;<br/><span class="kw">hence </span><a NAME="E9:80_1"/>
<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A6</font></span>: 
<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>  is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></span>;<br/>
<a NAME="E5:80"/>
<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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">FS</font>,<font color="Maroon" title="c4">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>  or <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">FS</font>,<font color="Maroon" title="c4">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:80_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">FS</font>,<font color="Maroon" title="c4">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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> <span class="kw"> such that </span><br/><a NAME="E3:80_2"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">A9</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> , <font color="Maroon" title="c10">d9</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c9">A9</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c11">Aq</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> , <font color="Maroon" title="c12">dq</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c11">Aq</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E5:80_2"/>
<font color="Maroon" title="c11">Aq</font>,<font color="Maroon" title="c12">dq</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c9">A9</font>,<font color="Maroon" title="c10">d9</font>
 <span class="kw">and </span><br/><a NAME="E6:80_2"/><span class="lab"><font color="Green" title="E64">A8</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">A9</font>,<font color="Maroon" title="c10">d9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:80_2"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">Aq</font>,<font color="Maroon" title="c12">dq</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<a NAME="E8:80_2"/>
<font color="Maroon" title="c9">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:80_2"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E9:80_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">A9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 
;<br/>
<a NAME="E10:80_2"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c9">A9</font>,<font color="Maroon" title="c9">A9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c9">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">FS</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E11:80_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A9</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">d9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">d9</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">FS</font>,<font color="Maroon" title="c4">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<a NAME="E12:80_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_2"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="txt" href="#E6:80_2"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:80_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">FS</font>,<font color="Maroon" title="c4">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:80_2"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="ref" href="partfun1.html#D3" title="PARTFUN1:def.3">PARTFUN1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:80"/><span class="kw">then </span>
 <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>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">FS</font>,<font color="Maroon" title="c4">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="ref" href="trees_2.html#T40" title="TREES_2:th.40">TREES_2:40</a></span>;<br/>
<a NAME="E7:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A10</font></span>: 
 ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <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="b2">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> where <font color="Olive" title="b2">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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">FS</font>,<font color="Maroon" title="c4">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D3" title="PARTFUN1:def.3">PARTFUN1:def 3</a></span>;<br/>
<a NAME="E8:80"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">X</font> = <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>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">LL</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">PP</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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> ;<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>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ;<br/>
<a NAME="E10:80"/><span class="lab"><font color="Green" title="E64">A11</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:80_3"/>
<span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> </span> } </span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> </span> } </span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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><br/>

<span class="kw">assume </span><a NAME="E1:80_3_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">J</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font><span class="kw"> such that </span><br/><a NAME="E3:80_3_1"/><span class="lab"><font color="Green" title="E65">A12</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c11">J</font>,<font color="Maroon" title="c11">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:80_3_1"/><span class="lab"><font color="Green" title="E66">A13</font></span>: 
<font color="Maroon" title="c11">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 ;<br/>
<a NAME="E5:80_3_1"/>
 ex <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 <font color="Maroon" title="c11">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_3_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:80_3_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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">by</span> <span class="lab"><a class="txt" href="#E3:80_3_1"><span class="lab"><font color="Green" title="E65">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:80_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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> <span class="kw"> such that </span><br/><a NAME="E4:80_3"/><span class="lab"><font color="Green" title="E65">A14</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><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="Maroon" title="c11">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<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="Maroon" title="c11">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>
<a NAME="E5:80_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:80_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">X</font> : <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_3"><span class="lab"><font color="Green" title="E65">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c10">FD</font> =  <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>  as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80"><span class="lab"><font color="Green" title="E63">A10</font></span></a></span>;<br/>
<a NAME="E12:80"/><span class="lab"><font color="Green" title="E65">A15</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</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">x</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E14:80"/><span class="lab"><font color="Green" title="E66">A16</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E15:80"/><span class="lab"><font color="Green" title="E67">A17</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E12:80"><span class="lab"><font color="Green" title="E65">A15</font></span></a>);<br/></span>
<a NAME="E16:80"/><span class="lab"><font color="Green" title="E68">A18</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:80_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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><br/>

<span class="kw">assume </span><a NAME="E1:80_4_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:80_4_1"/><span class="lab"><font color="Green" title="E69">A19</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">F</font>
 <span class="kw">and </span><br/><a NAME="E4:80_4_1"/><span class="lab"><font color="Green" title="E70">A20</font></span>: 
<font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">j</font> = <font color="Maroon" title="c13">j</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="txt" href="#E3:80_4_1"><span class="lab"><font color="Green" title="E69">A19</font></span></a></span>;<br/>
<a NAME="E6:80_4_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="txt" href="#E4:80_4_1"><span class="lab"><font color="Green" title="E70">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:80_4_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>  or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:80_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">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> <span class="kw"> such that </span><br/><a NAME="E4:80_4"/><span class="lab"><font color="Green" title="E69">A21</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ;<br/>
<a NAME="E5:80_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="txt" href="#E4:80_4"><span class="lab"><font color="Green" title="E69">A21</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:80_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:80"/>
<font color="Maroon" title="c11">F</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c11">F</font> or <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:80_5"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c12">x</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E66">A16</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">A1</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> , <font color="Maroon" title="c15">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c14">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c16">Aq1</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> , <font color="Maroon" title="c17">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c16">Aq1</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:80_5"/>
<font color="Maroon" title="c16">Aq1</font>,<font color="Maroon" title="c17">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c14">A1</font>,<font color="Maroon" title="c15">d1</font>
 <span class="kw">and </span><br/><a NAME="E5:80_5"/><span class="lab"><font color="Green" title="E69">A22</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">A1</font>,<font color="Maroon" title="c15">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:80_5"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">Aq1</font>,<font color="Maroon" title="c17">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<a NAME="E7:80_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_5"><span class="lab"><font color="Green" title="E69">A22</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:80_5"/>
<font color="Maroon" title="c11">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80"><span class="lab"><font color="Green" title="E67">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">F</font> = <font color="Maroon" title="c11">F</font> as   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E19:80"/><span class="lab"><font color="Green" title="E69">A23</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:80_6"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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><br/>

<span class="kw">assume </span><a NAME="E1:80_6_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:80_6_1"/><span class="lab"><font color="Green" title="E70">A24</font></span>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:80_6_1"/><span class="lab"><font color="Green" title="E71">A25</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:80_6_1"/><span class="lab"><font color="Green" title="E72">A26</font></span>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_6_1"><span class="lab"><font color="Green" title="E70">A24</font></span></a>, <a class="ref" href="funct_6.html#T59" title="FUNCT_6:th.59">FUNCT_6:59</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c14">j</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="txt" href="#E3:80_6_1"><span class="lab"><font color="Green" title="E70">A24</font></span></a>, <a class="ref" href="funct_6.html#T59" title="FUNCT_6:th.59">FUNCT_6:59</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">A1</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> , <font color="Maroon" title="c17">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c16">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c18">Aq1</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> , <font color="Maroon" title="c19">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c18">Aq1</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E8:80_6_1"/>
<font color="Maroon" title="c18">Aq1</font>,<font color="Maroon" title="c19">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c16">A1</font>,<font color="Maroon" title="c17">d1</font>
 <span class="kw">and </span><br/><a NAME="E9:80_6_1"/><span class="lab"><font color="Green" title="E73">A27</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">A1</font>,<font color="Maroon" title="c17">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:80_6_1"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">Aq1</font>,<font color="Maroon" title="c19">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<a NAME="E11:80_6_1"/><span class="lab"><font color="Green" title="E74">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:80_6_1"><span class="lab"><font color="Green" title="E73">A27</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E12:80_6_1"/><span class="lab"><font color="Green" title="E75">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:80_6_1"><span class="lab"><font color="Green" title="E73">A27</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<font color="Maroon" title="c13">x</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_6_1"><span class="lab"><font color="Green" title="E71">A25</font></span></a>, <a class="txt" href="#E5:80_6_1"><span class="lab"><font color="Green" title="E72">A26</font></span></a>, <a class="ref" href="funct_6.html#T22" title="FUNCT_6:th.22">FUNCT_6:22</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="txt" href="#E11:80_6_1"><span class="lab"><font color="Green" title="E74">A28</font></span></a></span>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><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="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<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="Maroon" title="c15">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:80_6_1"><span class="lab"><font color="Green" title="E75">A29</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:80_6_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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>  or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:80_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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>,<span class="p3">(<span class="default"><span class="p4">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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> <span class="kw"> such that </span><br/><a NAME="E4:80_6"/><span class="lab"><font color="Green" title="E70">A30</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><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="Maroon" title="c14">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<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="Maroon" title="c14">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">A1</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> , <font color="Maroon" title="c16">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c15">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c17">Aq1</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> , <font color="Maroon" title="c18">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c17">Aq1</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E6:80_6"/>
<font color="Maroon" title="c17">Aq1</font>,<font color="Maroon" title="c18">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c15">A1</font>,<font color="Maroon" title="c16">d1</font>
 <span class="kw">and </span><br/><a NAME="E7:80_6"/><span class="lab"><font color="Green" title="E71">A31</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">A1</font>,<font color="Maroon" title="c16">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:80_6"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">Aq1</font>,<font color="Maroon" title="c18">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<a NAME="E9:80_6"/><span class="lab"><font color="Green" title="E72">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_6"><span class="lab"><font color="Green" title="E71">A31</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E10:80_6"/>
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E11:80_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A33</font></span>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="ref" href="funct_6.html#T59" title="FUNCT_6:th.59">FUNCT_6:59</a></span>;<br/>
<a NAME="E12:80_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_6"><span class="lab"><font color="Green" title="E71">A31</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c13">x</font> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_6"><span class="lab"><font color="Green" title="E70">A30</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="txt" href="#E9:80_6"><span class="lab"><font color="Green" title="E72">A32</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="ref" href="funct_6.html#T22" title="FUNCT_6:th.22">FUNCT_6:22</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:80_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:80_6"><span class="lab"><font color="Green" title="E73">A33</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> implies <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:80_7"/><span class="lab"><font color="Green" title="E70">A34</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">and </span><br/><a NAME="E2:80_7"/><span class="lab"><font color="Green" title="E71">A35</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c15">k</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> <span class="kw"> such that </span><br/><a NAME="E4:80_7"/><span class="lab"><font color="Green" title="E72">A36</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_7"><span class="lab"><font color="Green" title="E70">A34</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">l</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> <span class="kw"> such that </span><br/><a NAME="E6:80_7"/><span class="lab"><font color="Green" title="E73">A37</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_7"><span class="lab"><font color="Green" title="E71">A35</font></span></a></span>;<br/><a NAME="E7:80_7"/>
( <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">l</font> or <font color="Maroon" title="c16">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> )
 ;<br/><a NAME="E8:80_7"/><span class="kw">then </span>
( <font color="Maroon" title="c13">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">y</font> or <font color="Maroon" title="c14">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_7"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="txt" href="#E6:80_7"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="ref" href="lattice5.html#T38" target="_self" title="LATTICE5:th.38">Th41</a></span>;<br/><span class="kw">hence </span><a NAME="E9:80_7"/>
<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E21:80"/><span class="kw">then </span>
<font color="Maroon" title="c7">X</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">FS</font>,<font color="Maroon" title="c4">FS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c12">F</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E19:80"><span class="lab"><font color="Green" title="E69">A23</font></span></a>, <a class="txt" href="#E10:80"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="ref" href="lattice5.html#T3" target="_self" title="LATTICE5:th.3">Th3</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">FD</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:80"><span class="lab"><font color="Green" title="E68">A18</font></span></a>, <a class="ref" href="lattice5.html#T1" target="_self" title="LATTICE5:th.1">Th1</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">FD</font> = <font color="Maroon" title="c10">FD</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c4">FS</font>,<font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E24:80"/><span class="lab"><font color="Green" title="E70">A38</font></span>: 
<font color="Maroon" title="c13">FD</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">FS</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice5.html#D5" target="_self" title="LATTICE5:def.5">LATTICE5:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c14">x</font>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c16">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:80_9"/><span class="lab"><font color="Green" title="E71">A39</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x1</font>
 <span class="kw">and </span><br/><a NAME="E3:80_9"/><span class="lab"><font color="Green" title="E72">A40</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">k</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> <span class="kw"> such that </span><br/><a NAME="E5:80_9"/><span class="lab"><font color="Green" title="E73">A41</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_9"><span class="lab"><font color="Green" title="E72">A40</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">A1</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> , <font color="Maroon" title="c19">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c18">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c20">Aq1</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> , <font color="Maroon" title="c21">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c20">Aq1</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:80_9"/>
<font color="Maroon" title="c20">Aq1</font>,<font color="Maroon" title="c21">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c18">A1</font>,<font color="Maroon" title="c19">d1</font>
 <span class="kw">and </span><br/><a NAME="E8:80_9"/><span class="lab"><font color="Green" title="E74">A42</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">A1</font>,<font color="Maroon" title="c19">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:80_9"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">Aq1</font>,<font color="Maroon" title="c21">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<a NAME="E10:80_9"/><span class="lab"><font color="Green" title="E75">A43</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_9"><span class="lab"><font color="Green" title="E71">A39</font></span></a>, <a class="txt" href="#E5:80_9"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="txt" href="#E8:80_9"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E11:80_9"/><span class="lab"><font color="Green" title="E76">A44</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80_9"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E12:80_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80_9"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E13:80_9"/><span class="kw">then </span>
<font color="Maroon" title="c19">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 
;<br/>
<a NAME="E14:80_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A45</font></span>: 
<font color="Maroon" title="c19">d1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c13">FD</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:80_9"/><span class="lab"><font color="Green" title="E78">A46</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">y1</font>
 <span class="kw">and </span><br/><a NAME="E17:80_9"/><span class="lab"><font color="Green" title="E79">A47</font></span>: 
<font color="Maroon" title="c22">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">l</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> <span class="kw"> such that </span><br/><a NAME="E19:80_9"/><span class="lab"><font color="Green" title="E80">A48</font></span>: 
<font color="Maroon" title="c22">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:80_9"><span class="lab"><font color="Green" title="E79">A47</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">A2</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> , <font color="Maroon" title="c25">d2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c24">A2</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c26">Aq2</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> , <font color="Maroon" title="c27">dq2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c26">Aq2</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E21:80_9"/>
<font color="Maroon" title="c26">Aq2</font>,<font color="Maroon" title="c27">dq2</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c24">A2</font>,<font color="Maroon" title="c25">d2</font>
 <span class="kw">and </span><br/><a NAME="E22:80_9"/><span class="lab"><font color="Green" title="E81">A49</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">A2</font>,<font color="Maroon" title="c25">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E23:80_9"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">Aq2</font>,<font color="Maroon" title="c27">dq2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<a NAME="E24:80_9"/><span class="lab"><font color="Green" title="E82">A50</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:80_9"><span class="lab"><font color="Green" title="E78">A46</font></span></a>, <a class="txt" href="#E19:80_9"><span class="lab"><font color="Green" title="E80">A48</font></span></a>, <a class="txt" href="#E22:80_9"><span class="lab"><font color="Green" title="E81">A49</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E25:80_9"/><span class="lab"><font color="Green" title="E83">A51</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:80_9"><span class="lab"><font color="Green" title="E81">A49</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E26:80_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">l</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">d2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:80_9"><span class="lab"><font color="Green" title="E81">A49</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E27:80_9"/><span class="kw">then </span>
<font color="Maroon" title="c25">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 
;<br/>
<a NAME="E28:80_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A52</font></span>: 
<font color="Maroon" title="c25">d2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c13">FD</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_9_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="E1:80_9_1"/>
( <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">l</font> or <font color="Maroon" title="c23">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_9_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_9_1_1"/>
<font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c14">x</font>)</span><br/><div class="add"><a NAME="E2:80_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:80_9"><span class="lab"><font color="Green" title="E76">A44</font></span></a>, <a class="txt" href="#E25:80_9"><span class="lab"><font color="Green" title="E83">A51</font></span></a>, <a class="ref" href="lattice5.html#T38" target="_self" title="LATTICE5:th.38">Th41</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">x9</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c29">y9</font> = <font color="Maroon" title="c15">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c24">A2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:80_9"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="txt" href="#E16:80_9"><span class="lab"><font color="Green" title="E78">A46</font></span></a>, <a class="txt" href="#E19:80_9"><span class="lab"><font color="Green" title="E80">A48</font></span></a>, <a class="txt" href="#E22:80_9"><span class="lab"><font color="Green" title="E81">A49</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/><a NAME="E4:80_9_1_1"/><span class="lab"><font color="Green" title="E85">A53</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c25">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c24">A2</font>,<font color="Maroon" title="c24">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>) = 
<font color="Maroon" title="c25">d2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">x9</font>,<font color="Maroon" title="c29">y9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:80_9"><span class="lab"><font color="Green" title="E84">A52</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c25">d2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c28">x9</font>,<font color="Maroon" title="c29">y9</font>)

<br/>.= 
<font color="Maroon" title="c25">d2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c29">y9</font>,<font color="Maroon" title="c28">x9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D5" target="_self" title="LATTICE5:def.5">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c13">FD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y9</font>,<font color="Maroon" title="c28">x9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:80_9"><span class="lab"><font color="Green" title="E84">A52</font></span></a>, <a class="txt" href="#E4:80_9_1_1"><span class="lab"><font color="Green" title="E85">A53</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c14">x</font>)

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_9_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_9_1_2"/>
<font color="Maroon" title="c23">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c14">x</font>)</span><br/><div class="add"><a NAME="E2:80_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:80_9"><span class="lab"><font color="Green" title="E76">A44</font></span></a>, <a class="txt" href="#E25:80_9"><span class="lab"><font color="Green" title="E83">A51</font></span></a>, <a class="ref" href="lattice5.html#T38" target="_self" title="LATTICE5:th.38">Th41</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">x9</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c29">y9</font> = <font color="Maroon" title="c15">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">A1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_9"><span class="lab"><font color="Green" title="E71">A39</font></span></a>, <a class="txt" href="#E5:80_9"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="txt" href="#E8:80_9"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="txt" href="#E24:80_9"><span class="lab"><font color="Green" title="E82">A50</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/><a NAME="E4:80_9_1_2"/><span class="lab"><font color="Green" title="E85">A54</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">A1</font>,<font color="Maroon" title="c18">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>) = 
<font color="Maroon" title="c19">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">x9</font>,<font color="Maroon" title="c29">y9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80_9"><span class="lab"><font color="Green" title="E77">A45</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c19">d1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c28">x9</font>,<font color="Maroon" title="c29">y9</font>)

<br/>.= 
<font color="Maroon" title="c19">d1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c29">y9</font>,<font color="Maroon" title="c28">x9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D5" target="_self" title="LATTICE5:def.5">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c13">FD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y9</font>,<font color="Maroon" title="c28">x9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80_9"><span class="lab"><font color="Green" title="E77">A45</font></span></a>, <a class="txt" href="#E4:80_9_1_2"><span class="lab"><font color="Green" title="E85">A54</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c14">x</font>)

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E25:80"/><span class="lab"><font color="Green" title="E71">A55</font></span>: 
<font color="Maroon" title="c13">FD</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</font>, <font color="Maroon" title="c16">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">FS</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice5.html#D7" target="_self" title="LATTICE5:def.7">LATTICE5:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c17">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:80_10"/><span class="lab"><font color="Green" title="E72">A56</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">x1</font>
 <span class="kw">and </span><br/><a NAME="E3:80_10"/><span class="lab"><font color="Green" title="E73">A57</font></span>: 
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">k</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> <span class="kw"> such that </span><br/><a NAME="E5:80_10"/><span class="lab"><font color="Green" title="E74">A58</font></span>: 
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_10"><span class="lab"><font color="Green" title="E73">A57</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">A1</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> , <font color="Maroon" title="c20">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c21">Aq1</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> , <font color="Maroon" title="c22">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c21">Aq1</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:80_10"/>
<font color="Maroon" title="c21">Aq1</font>,<font color="Maroon" title="c22">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c20">d1</font>
 <span class="kw">and </span><br/><a NAME="E8:80_10"/><span class="lab"><font color="Green" title="E75">A59</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c20">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:80_10"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">Aq1</font>,<font color="Maroon" title="c22">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<a NAME="E10:80_10"/><span class="lab"><font color="Green" title="E76">A60</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_10"><span class="lab"><font color="Green" title="E72">A56</font></span></a>, <a class="txt" href="#E5:80_10"><span class="lab"><font color="Green" title="E74">A58</font></span></a>, <a class="txt" href="#E8:80_10"><span class="lab"><font color="Green" title="E75">A59</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E11:80_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80_10"><span class="lab"><font color="Green" title="E75">A59</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E12:80_10"/><span class="kw">then </span>
<font color="Maroon" title="c20">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 
;<br/>
<a NAME="E13:80_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A61</font></span>: 
<font color="Maroon" title="c20">d1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c13">FD</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E14:80_10"/><span class="lab"><font color="Green" title="E78">A62</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c19">A1</font>,<font color="Maroon" title="c19">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:80_10"/><span class="lab"><font color="Green" title="E79">A63</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80_10"><span class="lab"><font color="Green" title="E75">A59</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:80_10"/><span class="lab"><font color="Green" title="E80">A64</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y1</font>
 <span class="kw">and </span><br/><a NAME="E18:80_10"/><span class="lab"><font color="Green" title="E81">A65</font></span>: 
<font color="Maroon" title="c23">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">l</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> <span class="kw"> such that </span><br/><a NAME="E20:80_10"/><span class="lab"><font color="Green" title="E82">A66</font></span>: 
<font color="Maroon" title="c23">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:80_10"><span class="lab"><font color="Green" title="E81">A65</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">A2</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> , <font color="Maroon" title="c26">d2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c25">A2</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c27">Aq2</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> , <font color="Maroon" title="c28">dq2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c27">Aq2</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E22:80_10"/>
<font color="Maroon" title="c27">Aq2</font>,<font color="Maroon" title="c28">dq2</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c25">A2</font>,<font color="Maroon" title="c26">d2</font>
 <span class="kw">and </span><br/><a NAME="E23:80_10"/><span class="lab"><font color="Green" title="E83">A67</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">A2</font>,<font color="Maroon" title="c26">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E24:80_10"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">Aq2</font>,<font color="Maroon" title="c28">dq2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<a NAME="E25:80_10"/><span class="lab"><font color="Green" title="E84">A68</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:80_10"><span class="lab"><font color="Green" title="E80">A64</font></span></a>, <a class="txt" href="#E20:80_10"><span class="lab"><font color="Green" title="E82">A66</font></span></a>, <a class="txt" href="#E23:80_10"><span class="lab"><font color="Green" title="E83">A67</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E26:80_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">d2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:80_10"><span class="lab"><font color="Green" title="E83">A67</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E27:80_10"/><span class="kw">then </span>
<font color="Maroon" title="c26">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 
;<br/>
<a NAME="E28:80_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A69</font></span>: 
<font color="Maroon" title="c26">d2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c13">FD</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E29:80_10"/><span class="lab"><font color="Green" title="E86">A70</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c26">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c25">A2</font>,<font color="Maroon" title="c25">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E30:80_10"/><span class="lab"><font color="Green" title="E87">A71</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:80_10"><span class="lab"><font color="Green" title="E83">A67</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c29">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E32:80_10"/><span class="lab"><font color="Green" title="E88">A72</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">z1</font>
 <span class="kw">and </span><br/><a NAME="E33:80_10"/><span class="lab"><font color="Green" title="E89">A73</font></span>: 
<font color="Maroon" title="c29">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c30">n</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> <span class="kw"> such that </span><br/><a NAME="E35:80_10"/><span class="lab"><font color="Green" title="E90">A74</font></span>: 
<font color="Maroon" title="c29">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:80_10"><span class="lab"><font color="Green" title="E89">A73</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c31">A3</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> , <font color="Maroon" title="c32">d3</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c31">A3</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c33">Aq3</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> , <font color="Maroon" title="c34">dq3</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c33">Aq3</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E37:80_10"/>
<font color="Maroon" title="c33">Aq3</font>,<font color="Maroon" title="c34">dq3</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c31">A3</font>,<font color="Maroon" title="c32">d3</font>
 <span class="kw">and </span><br/><a NAME="E38:80_10"/><span class="lab"><font color="Green" title="E91">A75</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">A3</font>,<font color="Maroon" title="c32">d3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E39:80_10"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">Aq3</font>,<font color="Maroon" title="c34">dq3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<a NAME="E40:80_10"/><span class="lab"><font color="Green" title="E92">A76</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:80_10"><span class="lab"><font color="Green" title="E88">A72</font></span></a>, <a class="txt" href="#E35:80_10"><span class="lab"><font color="Green" title="E90">A74</font></span></a>, <a class="txt" href="#E38:80_10"><span class="lab"><font color="Green" title="E91">A75</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E41:80_10"/><span class="lab"><font color="Green" title="E93">A77</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">A3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:80_10"><span class="lab"><font color="Green" title="E91">A75</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E42:80_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">d3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:80_10"><span class="lab"><font color="Green" title="E91">A75</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E43:80_10"/><span class="kw">then </span>
<font color="Maroon" title="c32">d3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 
;<br/>
<a NAME="E44:80_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A78</font></span>: 
<font color="Maroon" title="c32">d3</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c13">FD</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E45:80_10"/><span class="lab"><font color="Green" title="E95">A79</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c32">d3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c31">A3</font>,<font color="Maroon" title="c31">A3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_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="E1:80_10_1"/>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">l</font> or <font color="Maroon" title="c24">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_10_1_1"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)</span><br/><div class="add"><a NAME="E2:80_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A80</font></span>: 
<font color="Maroon" title="c19">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80_10"><span class="lab"><font color="Green" title="E79">A63</font></span></a>, <a class="txt" href="#E30:80_10"><span class="lab"><font color="Green" title="E87">A71</font></span></a>, <a class="ref" href="lattice5.html#T38" target="_self" title="LATTICE5:th.38">Th41</a></span>;<br/><span class="kw">thus </span><a NAME="E3:80_10_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_10_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:80_10_1_1_1_1"/>
( <font color="Maroon" title="c24">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">n</font> or <font color="Maroon" title="c30">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">l</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_10_1_1_1_1_1"/>
<font color="Maroon" title="c24">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)</span><br/><div class="add"><a NAME="E2:80_10_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A81</font></span>: 
<font color="Maroon" title="c25">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c31">A3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:80_10"><span class="lab"><font color="Green" title="E87">A71</font></span></a>, <a class="txt" href="#E41:80_10"><span class="lab"><font color="Green" title="E93">A77</font></span></a>, <a class="ref" href="lattice5.html#T38" target="_self" title="LATTICE5:th.38">Th41</a></span>;<br/><a NAME="E3:80_10_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c31">A3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_10_1_1"><span class="lab"><font color="Green" title="E96">A80</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">x9</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c36">y9</font> = <font color="Maroon" title="c15">y</font>, <font color="Maroon" title="c37">z9</font> = <font color="Maroon" title="c16">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c31">A3</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:80_10"><span class="lab"><font color="Green" title="E76">A60</font></span></a>, <a class="txt" href="#E25:80_10"><span class="lab"><font color="Green" title="E84">A68</font></span></a>, <a class="txt" href="#E32:80_10"><span class="lab"><font color="Green" title="E88">A72</font></span></a>, <a class="txt" href="#E35:80_10"><span class="lab"><font color="Green" title="E90">A74</font></span></a>, <a class="txt" href="#E38:80_10"><span class="lab"><font color="Green" title="E91">A75</font></span></a>, <a class="txt" href="#E2:80_10_1_1_1_1_1"><span class="lab"><font color="Green" title="E97">A81</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E98">A82</font></span>: <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>) = 
<font color="Maroon" title="c32">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">y9</font>,<font color="Maroon" title="c37">z9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:80_10"><span class="lab"><font color="Green" title="E94">A78</font></span></a>, <a class="txt" href="#E45:80_10"><span class="lab"><font color="Green" title="E95">A79</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c32">d3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c36">y9</font>,<font color="Maroon" title="c37">z9</font>)

;<br/><span class="lab"><font color="Green" title="E99">A83</font></span>: <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>) = 
<font color="Maroon" title="c32">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c37">z9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:80_10"><span class="lab"><font color="Green" title="E94">A78</font></span></a>, <a class="txt" href="#E45:80_10"><span class="lab"><font color="Green" title="E95">A79</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c32">d3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c37">z9</font>)

;<br/><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>) = 
<font color="Maroon" title="c32">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c36">y9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:80_10"><span class="lab"><font color="Green" title="E94">A78</font></span></a>, <a class="txt" href="#E45:80_10"><span class="lab"><font color="Green" title="E95">A79</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c32">d3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c36">y9</font>)

;<br/><span class="kw">hence </span><a NAME="E8:80_10_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_10_1_1_1_1_1"><span class="lab"><font color="Green" title="E98">A82</font></span></a>, <a class="txt" href="#E6:80_10_1_1_1_1_1"><span class="lab"><font color="Green" title="E99">A83</font></span></a>, <a class="ref" href="lattice5.html#D7" target="_self" title="LATTICE5:def.7">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_10_1_1_1_1_2"/>
<font color="Maroon" title="c30">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)</span><br/><div class="add"><a NAME="E2:80_10_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c31">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:80_10"><span class="lab"><font color="Green" title="E87">A71</font></span></a>, <a class="txt" href="#E41:80_10"><span class="lab"><font color="Green" title="E93">A77</font></span></a>, <a class="ref" href="lattice5.html#T38" target="_self" title="LATTICE5:th.38">Th41</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">x9</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c36">y9</font> = <font color="Maroon" title="c15">y</font>, <font color="Maroon" title="c37">z9</font> = <font color="Maroon" title="c16">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c25">A2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:80_10"><span class="lab"><font color="Green" title="E76">A60</font></span></a>, <a class="txt" href="#E17:80_10"><span class="lab"><font color="Green" title="E80">A64</font></span></a>, <a class="txt" href="#E20:80_10"><span class="lab"><font color="Green" title="E82">A66</font></span></a>, <a class="txt" href="#E23:80_10"><span class="lab"><font color="Green" title="E83">A67</font></span></a>, <a class="txt" href="#E40:80_10"><span class="lab"><font color="Green" title="E92">A76</font></span></a>, <a class="txt" href="#E2:80_10_1_1"><span class="lab"><font color="Green" title="E96">A80</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E97">A84</font></span>: <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>) = 
<font color="Maroon" title="c26">d2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">y9</font>,<font color="Maroon" title="c37">z9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:80_10"><span class="lab"><font color="Green" title="E85">A69</font></span></a>, <a class="txt" href="#E29:80_10"><span class="lab"><font color="Green" title="E86">A70</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c26">d2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c36">y9</font>,<font color="Maroon" title="c37">z9</font>)

;<br/><span class="lab"><font color="Green" title="E98">A85</font></span>: <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>) = 
<font color="Maroon" title="c26">d2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c37">z9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:80_10"><span class="lab"><font color="Green" title="E85">A69</font></span></a>, <a class="txt" href="#E29:80_10"><span class="lab"><font color="Green" title="E86">A70</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c26">d2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c37">z9</font>)

;<br/><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>) = 
<font color="Maroon" title="c26">d2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c36">y9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:80_10"><span class="lab"><font color="Green" title="E85">A69</font></span></a>, <a class="txt" href="#E29:80_10"><span class="lab"><font color="Green" title="E86">A70</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c26">d2</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c36">y9</font>)

;<br/><span class="kw">hence </span><a NAME="E7:80_10_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_10_1_1_1_1_2"><span class="lab"><font color="Green" title="E97">A84</font></span></a>, <a class="txt" href="#E5:80_10_1_1_1_1_2"><span class="lab"><font color="Green" title="E98">A85</font></span></a>, <a class="ref" href="lattice5.html#D7" target="_self" title="LATTICE5:def.7">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_10_1_2"/>
<font color="Maroon" title="c24">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)</span><br/><div class="add"><a NAME="E2:80_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A86</font></span>: 
<font color="Maroon" title="c25">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80_10"><span class="lab"><font color="Green" title="E79">A63</font></span></a>, <a class="txt" href="#E30:80_10"><span class="lab"><font color="Green" title="E87">A71</font></span></a>, <a class="ref" href="lattice5.html#T38" target="_self" title="LATTICE5:th.38">Th41</a></span>;<br/><span class="kw">thus </span><a NAME="E3:80_10_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_10_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:80_10_1_2_1_1"/>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">n</font> or <font color="Maroon" title="c30">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_10_1_2_1_1_1"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)</span><br/><div class="add"><a NAME="E2:80_10_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A87</font></span>: 
<font color="Maroon" title="c19">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c31">A3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80_10"><span class="lab"><font color="Green" title="E79">A63</font></span></a>, <a class="txt" href="#E41:80_10"><span class="lab"><font color="Green" title="E93">A77</font></span></a>, <a class="ref" href="lattice5.html#T38" target="_self" title="LATTICE5:th.38">Th41</a></span>;<br/><a NAME="E3:80_10_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c31">A3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_10_1_2"><span class="lab"><font color="Green" title="E96">A86</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">x9</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c36">y9</font> = <font color="Maroon" title="c15">y</font>, <font color="Maroon" title="c37">z9</font> = <font color="Maroon" title="c16">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c31">A3</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:80_10"><span class="lab"><font color="Green" title="E76">A60</font></span></a>, <a class="txt" href="#E25:80_10"><span class="lab"><font color="Green" title="E84">A68</font></span></a>, <a class="txt" href="#E32:80_10"><span class="lab"><font color="Green" title="E88">A72</font></span></a>, <a class="txt" href="#E35:80_10"><span class="lab"><font color="Green" title="E90">A74</font></span></a>, <a class="txt" href="#E38:80_10"><span class="lab"><font color="Green" title="E91">A75</font></span></a>, <a class="txt" href="#E2:80_10_1_2_1_1_1"><span class="lab"><font color="Green" title="E97">A87</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E98">A88</font></span>: <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>) = 
<font color="Maroon" title="c32">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">y9</font>,<font color="Maroon" title="c37">z9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:80_10"><span class="lab"><font color="Green" title="E94">A78</font></span></a>, <a class="txt" href="#E45:80_10"><span class="lab"><font color="Green" title="E95">A79</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c32">d3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c36">y9</font>,<font color="Maroon" title="c37">z9</font>)

;<br/><span class="lab"><font color="Green" title="E99">A89</font></span>: <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>) = 
<font color="Maroon" title="c32">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c37">z9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:80_10"><span class="lab"><font color="Green" title="E94">A78</font></span></a>, <a class="txt" href="#E45:80_10"><span class="lab"><font color="Green" title="E95">A79</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c32">d3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c37">z9</font>)

;<br/><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>) = 
<font color="Maroon" title="c32">d3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c36">y9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:80_10"><span class="lab"><font color="Green" title="E94">A78</font></span></a>, <a class="txt" href="#E45:80_10"><span class="lab"><font color="Green" title="E95">A79</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c32">d3</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c36">y9</font>)

;<br/><span class="kw">hence </span><a NAME="E8:80_10_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_10_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A88</font></span></a>, <a class="txt" href="#E6:80_10_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A89</font></span></a>, <a class="ref" href="lattice5.html#D7" target="_self" title="LATTICE5:def.7">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_10_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_10_1_2_1_1_2"/>
<font color="Maroon" title="c30">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)</span><br/><div class="add"><a NAME="E2:80_10_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c31">A3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80_10"><span class="lab"><font color="Green" title="E79">A63</font></span></a>, <a class="txt" href="#E41:80_10"><span class="lab"><font color="Green" title="E93">A77</font></span></a>, <a class="ref" href="lattice5.html#T38" target="_self" title="LATTICE5:th.38">Th41</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">x9</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c36">y9</font> = <font color="Maroon" title="c15">y</font>, <font color="Maroon" title="c37">z9</font> = <font color="Maroon" title="c16">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">A1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_10"><span class="lab"><font color="Green" title="E72">A56</font></span></a>, <a class="txt" href="#E5:80_10"><span class="lab"><font color="Green" title="E74">A58</font></span></a>, <a class="txt" href="#E8:80_10"><span class="lab"><font color="Green" title="E75">A59</font></span></a>, <a class="txt" href="#E25:80_10"><span class="lab"><font color="Green" title="E84">A68</font></span></a>, <a class="txt" href="#E40:80_10"><span class="lab"><font color="Green" title="E92">A76</font></span></a>, <a class="txt" href="#E2:80_10_1_2"><span class="lab"><font color="Green" title="E96">A86</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E97">A90</font></span>: <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>) = 
<font color="Maroon" title="c20">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">y9</font>,<font color="Maroon" title="c37">z9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:80_10"><span class="lab"><font color="Green" title="E77">A61</font></span></a>, <a class="txt" href="#E14:80_10"><span class="lab"><font color="Green" title="E78">A62</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c20">d1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c36">y9</font>,<font color="Maroon" title="c37">z9</font>)

;<br/><span class="lab"><font color="Green" title="E98">A91</font></span>: <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>) = 
<font color="Maroon" title="c20">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c37">z9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:80_10"><span class="lab"><font color="Green" title="E77">A61</font></span></a>, <a class="txt" href="#E14:80_10"><span class="lab"><font color="Green" title="E78">A62</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c20">d1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c37">z9</font>)

;<br/><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>) = 
<font color="Maroon" title="c20">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c36">y9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:80_10"><span class="lab"><font color="Green" title="E77">A61</font></span></a>, <a class="txt" href="#E14:80_10"><span class="lab"><font color="Green" title="E78">A62</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c20">d1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c35">x9</font>,<font color="Maroon" title="c36">y9</font>)

;<br/><span class="kw">hence </span><a NAME="E7:80_10_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>)</span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c16">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_10_1_2_1_1_2"><span class="lab"><font color="Green" title="E97">A90</font></span></a>, <a class="txt" href="#E5:80_10_1_2_1_1_2"><span class="lab"><font color="Green" title="E98">A91</font></span></a>, <a class="ref" href="lattice5.html#D7" target="_self" title="LATTICE5:def.7">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E26:80"/>
<font color="Maroon" title="c13">FD</font> is  <a href="lattice5.html#V2" title="LATTICE5:attr.2">zeroed</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">FS</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">LATTICE5:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c14">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:80_11"/><span class="lab"><font color="Green" title="E72">A92</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">y</font>
 <span class="kw">and </span><br/><a NAME="E3:80_11"/><span class="lab"><font color="Green" title="E73">A93</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">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> <span class="kw"> such that </span><br/><a NAME="E5:80_11"/><span class="lab"><font color="Green" title="E74">A94</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_11"><span class="lab"><font color="Green" title="E73">A93</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">A1</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> , <font color="Maroon" title="c18">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c19">Aq1</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> , <font color="Maroon" title="c20">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c19">Aq1</font>,<font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:80_11"/>
<font color="Maroon" title="c19">Aq1</font>,<font color="Maroon" title="c20">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c18">d1</font>
 <span class="kw">and </span><br/><a NAME="E8:80_11"/><span class="lab"><font color="Green" title="E75">A95</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c18">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:80_11"/>
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">Aq1</font>,<font color="Maroon" title="c20">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D20" target="_self" title="LATTICE5:def.20">Def21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">x9</font> = <font color="Maroon" title="c14">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">A1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_11"><span class="lab"><font color="Green" title="E72">A92</font></span></a>, <a class="txt" href="#E5:80_11"><span class="lab"><font color="Green" title="E74">A94</font></span></a>, <a class="txt" href="#E8:80_11"><span class="lab"><font color="Green" title="E75">A95</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>;<br/>
<a NAME="E11:80_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80_11"><span class="lab"><font color="Green" title="E75">A95</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/>
<a NAME="E12:80_11"/><span class="kw">then </span>
<font color="Maroon" title="c18">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 
;<br/>
<a NAME="E13:80_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A96</font></span>: 
<font color="Maroon" title="c18">d1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c13">FD</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</a></span>;<br/>
<a NAME="E14:80_11"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">A1</font>,<font color="Maroon" title="c17">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c13">FD</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c14">x</font>) = 
<font color="Maroon" title="c18">d1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">x9</font>,<font color="Maroon" title="c21">x9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:80_11"><span class="lab"><font color="Green" title="E76">A96</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c18">d1</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c21">x9</font>,<font color="Maroon" title="c21">x9</font>)

<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E27:80"/>
 <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>  is   <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:80"><span class="lab"><font color="Green" title="E70">A38</font></span></a>, <a class="txt" href="#E25:80"><span class="lab"><font color="Green" title="E71">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
