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<b>let </b><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>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be    <a href="lattice5.html#M2" title="LATTICE5:mode.2">ExtensionSeq</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>, <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font>;<br/><b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c4">FD</font> be   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c3">FS</font>,<font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font>;<br/><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green" title="E62">A1</font></i>: 
( <font color="Maroon" title="c3">FS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c4">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  )
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c8">b</font>
 ;<br/>

<a NAME="E3:82"/>
<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/>
<b>then reconsider </b><font color="Maroon" title="c9">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/>
<b>consider </b><font color="Maroon" title="c10">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:82"/><i><font color="Green" title="E64">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x1</font> &amp; <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">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> <b> such that </b><br/><a NAME="E8:82"/><i><font color="Green" title="E65">A4</font></i>: 
<font color="Maroon" title="c10">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="c11">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E6:82"><i><font color="Green" title="E64">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">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="c13">d1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c12">A1</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c14">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="c15">dq1</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c14">Aq1</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E10:82"/><i><font color="Green" title="E66">A5</font></i>: 
<font color="Maroon" title="c14">Aq1</font>,<font color="Maroon" title="c15">dq1</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c12">A1</font>,<font color="Maroon" title="c13">d1</font>
 <b>and </b><br/><a NAME="E11:82"/><i><font color="Green" title="E67">A6</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c12">A1</font>,<font color="Maroon" title="c13">d1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <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="c11">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="c14">Aq1</font>,<font color="Maroon" title="c15">dq1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E12:82"/><i><font color="Green" title="E68">A7</font></i>: 
<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">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="c13">d1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E11:82"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E13:82"/><i><font color="Green" title="E69">A8</font></i>: 
<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">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="c12">A1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E11:82"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E14:82"/><i><font color="Green" title="E70">A9</font></i>: 
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c14">Aq1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E11:82"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E15:82"/><i><font color="Green" title="E71">A10</font></i>: 
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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">dq1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E11:82"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E16:82"/>
<font color="Maroon" title="c13">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> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E12:82"><i><font color="Green" title="E68">A7</font></i></a></i>;<br/>
<a NAME="E17:82"/><b>then </b><i><font color="Green" title="E72">A11</font></i>: 
<font color="Maroon" title="c13">d1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E18:82"/>
<font color="Maroon" title="c15">dq1</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> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E15:82"><i><font color="Green" title="E71">A10</font></i></a></i>;<br/>
<a NAME="E19:82"/><b>then </b><i><font color="Green" title="E73">A12</font></i>: 
<font color="Maroon" title="c15">dq1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E20:82"/>
<font color="Maroon" title="c14">Aq1</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> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E14:82"><i><font color="Green" title="E70">A9</font></i></a></i>;<br/>
<a NAME="E21:82"/><b>then </b><i><font color="Green" title="E74">A13</font></i>: 
<font color="Maroon" title="c14">Aq1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E22:82"/><i><font color="Green" title="E75">A14</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E6:82"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="lattice5.html#E8:82"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="lattice5.html#E11:82"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:82"/><i><font color="Green" title="E76">A15</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">y1</font> &amp; <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">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> <b> such that </b><br/><a NAME="E26:82"/><i><font color="Green" title="E77">A16</font></i>: 
<font color="Maroon" title="c16">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="c17">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E24:82"><i><font color="Green" title="E76">A15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">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="c19">d2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c18">A2</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c20">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="c21">dq2</font> being   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c20">Aq2</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E28:82"/><i><font color="Green" title="E78">A17</font></i>: 
<font color="Maroon" title="c20">Aq2</font>,<font color="Maroon" title="c21">dq2</font> <a href="lattice5.html#R2" title="LATTICE5:pred.2">is_extension_of</a> <font color="Maroon" title="c18">A2</font>,<font color="Maroon" title="c19">d2</font>
 <b>and </b><br/><a NAME="E29:82"/><i><font color="Green" title="E79">A18</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c18">A2</font>,<font color="Maroon" title="c19">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <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">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="c20">Aq2</font>,<font color="Maroon" title="c21">dq2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self" title="LATTICE5:def.21">Def21</a></i>;<br/>
<a NAME="E30:82"/><i><font color="Green" title="E80">A19</font></i>: 
<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">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="c19">d2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E29:82"><i><font color="Green" title="E79">A18</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E31:82"/><i><font color="Green" title="E81">A20</font></i>: 
<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">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="c18">A2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E29:82"><i><font color="Green" title="E79">A18</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E32:82"/><i><font color="Green" title="E82">A21</font></i>: 
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c17">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c20">Aq2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E29:82"><i><font color="Green" title="E79">A18</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<a NAME="E33:82"/><i><font color="Green" title="E83">A22</font></i>: 
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c17">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c21">dq2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E29:82"><i><font color="Green" title="E79">A18</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E34:82"/>
<font color="Maroon" title="c19">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> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E30:82"><i><font color="Green" title="E80">A19</font></i></a></i>;<br/>
<a NAME="E35:82"/><b>then </b><i><font color="Green" title="E84">A23</font></i>: 
<font color="Maroon" title="c19">d2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E36:82"/>
<font color="Maroon" title="c21">dq2</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> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E33:82"><i><font color="Green" title="E83">A22</font></i></a></i>;<br/>
<a NAME="E37:82"/><b>then </b><i><font color="Green" title="E85">A24</font></i>: 
<font color="Maroon" title="c21">dq2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E38:82"/>
<font color="Maroon" title="c20">Aq2</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> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E32:82"><i><font color="Green" title="E82">A21</font></i></a></i>;<br/>
<a NAME="E39:82"/><b>then </b><i><font color="Green" title="E86">A25</font></i>: 
<font color="Maroon" title="c20">Aq2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E40:82"/><i><font color="Green" title="E87">A26</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E24:82"><i><font color="Green" title="E76">A15</font></i></a>, <a class="txt" href="lattice5.html#E26:82"><i><font color="Green" title="E77">A16</font></i></a>, <a class="txt" href="lattice5.html#E29:82"><i><font color="Green" title="E79">A18</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1"/>
( <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font> or <font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><b>suppose </b></a><a NAME="E1:82_1_1"/>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font>
 ;<br/><div class="add"><a NAME="E2:82_1_1"/><b>then </b>
<font color="Maroon" title="c12">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">A2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E13:82"><i><font color="Green" title="E69">A8</font></i></a>, <a class="txt" href="lattice5.html#E31:82"><i><font color="Green" title="E81">A20</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c23">y'</font> = <font color="Maroon" title="c6">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">A2</font> <b>by </b><i><a class="txt" href="lattice5.html#E22:82"><i><font color="Green" title="E75">A14</font></i></a>, <a class="txt" href="lattice5.html#E24:82"><i><font color="Green" title="E76">A15</font></i></a>, <a class="txt" href="lattice5.html#E26:82"><i><font color="Green" title="E77">A16</font></i></a>, <a class="txt" href="lattice5.html#E29:82"><i><font color="Green" title="E79">A18</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/><a NAME="E4:82_1_1"/><i><font color="Green" title="E88">A27</font></i>: 
( <font color="Maroon" title="c22">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A2</font> &amp; <font color="Maroon" title="c23">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A2</font> )
 ;<br/><a NAME="E5:82_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">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="c18">A2</font>,<font color="Maroon" title="c18">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> = 
<font color="Maroon" title="c19">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="c22">x'</font>,<font color="Maroon" title="c23">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E35:82"><i><font color="Green" title="E84">A23</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c19">d2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c22">x'</font>,<font color="Maroon" title="c23">y'</font>

;<br/><b>then consider </b><font color="Maroon" title="c24">z1</font>, <font color="Maroon" title="c25">z2</font>, <font color="Maroon" title="c26">z3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c20">Aq2</font><b> such that </b><br/><a NAME="E8:82_1_1"/><i><font color="Green" title="E89">A28</font></i>: 
<font color="Maroon" title="c21">dq2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c24">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <b>and </b><br/><a NAME="E9:82_1_1"/><i><font color="Green" title="E90">A29</font></i>: 
<font color="Maroon" title="c21">dq2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c25">z2</font>,<font color="Maroon" title="c26">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <b>and </b><br/><a NAME="E10:82_1_1"/><i><font color="Green" title="E91">A30</font></i>: 
<font color="Maroon" title="c21">dq2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c24">z1</font>,<font color="Maroon" title="c25">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <b>and </b><br/><a NAME="E11:82_1_1"/><i><font color="Green" title="E92">A31</font></i>: 
<font color="Maroon" title="c21">dq2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">z3</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:82"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="lattice5.html#E28:82"><i><font color="Green" title="E78">A17</font></i></a>, <a class="ref" href="lattice5.html#T40" target="_self" title="LATTICE5:th.40">Th40</a></i>;<br/><a NAME="E12:82_1_1"/>
( <font color="Maroon" title="c24">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Aq2</font> &amp; <font color="Maroon" title="c25">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Aq2</font> &amp; <font color="Maroon" title="c26">z3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Aq2</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c27">z1'</font> = <font color="Maroon" title="c24">z1</font>, <font color="Maroon" title="c28">z2'</font> = <font color="Maroon" title="c25">z2</font>, <font color="Maroon" title="c29">z3'</font> = <font color="Maroon" title="c26">z3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> <b>by </b><i><a class="txt" href="lattice5.html#E39:82"><i><font color="Green" title="E86">A25</font></i></a></i>;<br/><a NAME="E14:82_1_1"/>
<font color="Maroon" title="c18">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">Aq2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E31:82"><i><font color="Green" title="E81">A20</font></i></a>, <a class="txt" href="lattice5.html#E32:82"><i><font color="Green" title="E82">A21</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">x''</font> = <font color="Maroon" title="c22">x'</font>, <font color="Maroon" title="c31">y''</font> = <font color="Maroon" title="c23">y'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c20">Aq2</font> <b>by </b><i><a class="txt" href="lattice5.html#E4:82_1_1"><i><font color="Green" title="E88">A27</font></i></a></i>;<br/><a NAME="E16:82_1_1"/><i><font color="Green" title="E93">A32</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">dq2</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="c20">Aq2</font>,<font color="Maroon" title="c20">Aq2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c27">z1'</font>
;<br/><b>take </b>
<font color="Maroon" title="c28">z2'</font>
;<br/><b>take </b>
<font color="Maroon" title="c29">z3'</font>
;<br/><b>thus </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c27">z1'</font> = 
<font color="Maroon" title="c21">dq2</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="c30">x''</font>,<font color="Maroon" title="c24">z1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green" title="E85">A24</font></i></a>, <a class="txt" href="lattice5.html#E16:82_1_1"><i><font color="Green" title="E93">A32</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E8:82_1_1"><i><font color="Green" title="E89">A28</font></i></a></i>
;<br/><b>thus </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c28">z2'</font>,<font color="Maroon" title="c29">z3'</font> = 
<font color="Maroon" title="c21">dq2</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="c25">z2</font>,<font color="Maroon" title="c26">z3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green" title="E85">A24</font></i></a>, <a class="txt" href="lattice5.html#E16:82_1_1"><i><font color="Green" title="E93">A32</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E9:82_1_1"><i><font color="Green" title="E90">A29</font></i></a></i>
;<br/><b>thus </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c27">z1'</font>,<font color="Maroon" title="c28">z2'</font> = 
<font color="Maroon" title="c21">dq2</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="c24">z1</font>,<font color="Maroon" title="c25">z2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green" title="E85">A24</font></i></a>, <a class="txt" href="lattice5.html#E16:82_1_1"><i><font color="Green" title="E93">A32</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E10:82_1_1"><i><font color="Green" title="E91">A30</font></i></a></i>
;<br/><b>thus </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c29">z3'</font>,<font color="Maroon" title="c6">y</font> = 
<font color="Maroon" title="c21">dq2</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="c26">z3</font>,<font color="Maroon" title="c31">y''</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green" title="E85">A24</font></i></a>, <a class="txt" href="lattice5.html#E16:82_1_1"><i><font color="Green" title="E93">A32</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E11:82_1_1"><i><font color="Green" title="E92">A31</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><b>suppose </b></a><a NAME="E1:82_1_2"/>
<font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 ;<br/><div class="add"><a NAME="E2:82_1_2"/><b>then </b>
<font color="Maroon" title="c18">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E13:82"><i><font color="Green" title="E69">A8</font></i></a>, <a class="txt" href="lattice5.html#E31:82"><i><font color="Green" title="E81">A20</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c23">y'</font> = <font color="Maroon" title="c6">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">A1</font> <b>by </b><i><a class="txt" href="lattice5.html#E6:82"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="lattice5.html#E8:82"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="lattice5.html#E11:82"><i><font color="Green" title="E67">A6</font></i></a>, <a class="txt" href="lattice5.html#E40:82"><i><font color="Green" title="E87">A26</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>;<br/><a NAME="E4:82_1_2"/><i><font color="Green" title="E88">A33</font></i>: 
( <font color="Maroon" title="c22">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A1</font> &amp; <font color="Maroon" title="c23">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A1</font> )
 ;<br/><a NAME="E5:82_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">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="c12">A1</font>,<font color="Maroon" title="c12">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> = 
<font color="Maroon" title="c13">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="c22">x'</font>,<font color="Maroon" title="c23">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E17:82"><i><font color="Green" title="E72">A11</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c13">d1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c22">x'</font>,<font color="Maroon" title="c23">y'</font>

;<br/><b>then consider </b><font color="Maroon" title="c24">z1</font>, <font color="Maroon" title="c25">z2</font>, <font color="Maroon" title="c26">z3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Aq1</font><b> such that </b><br/><a NAME="E8:82_1_2"/><i><font color="Green" title="E89">A34</font></i>: 
<font color="Maroon" title="c15">dq1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c24">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <b>and </b><br/><a NAME="E9:82_1_2"/><i><font color="Green" title="E90">A35</font></i>: 
<font color="Maroon" title="c15">dq1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c25">z2</font>,<font color="Maroon" title="c26">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <b>and </b><br/><a NAME="E10:82_1_2"/><i><font color="Green" title="E91">A36</font></i>: 
<font color="Maroon" title="c15">dq1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c24">z1</font>,<font color="Maroon" title="c25">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <b>and </b><br/><a NAME="E11:82_1_2"/><i><font color="Green" title="E92">A37</font></i>: 
<font color="Maroon" title="c15">dq1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">z3</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:82"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="lattice5.html#E10:82"><i><font color="Green" title="E66">A5</font></i></a>, <a class="ref" href="lattice5.html#T40" target="_self" title="LATTICE5:th.40">Th40</a></i>;<br/><a NAME="E12:82_1_2"/>
( <font color="Maroon" title="c24">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Aq1</font> &amp; <font color="Maroon" title="c25">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Aq1</font> &amp; <font color="Maroon" title="c26">z3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Aq1</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c27">z1'</font> = <font color="Maroon" title="c24">z1</font>, <font color="Maroon" title="c28">z2'</font> = <font color="Maroon" title="c25">z2</font>, <font color="Maroon" title="c29">z3'</font> = <font color="Maroon" title="c26">z3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> <b>by </b><i><a class="txt" href="lattice5.html#E21:82"><i><font color="Green" title="E74">A13</font></i></a></i>;<br/><a NAME="E14:82_1_2"/>
<font color="Maroon" title="c12">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">Aq1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E13:82"><i><font color="Green" title="E69">A8</font></i></a>, <a class="txt" href="lattice5.html#E14:82"><i><font color="Green" title="E70">A9</font></i></a>, <a class="ref" href="lattice5.html#T41" target="_self" title="LATTICE5:th.41">Th41</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">x''</font> = <font color="Maroon" title="c22">x'</font>, <font color="Maroon" title="c31">y''</font> = <font color="Maroon" title="c23">y'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Aq1</font> <b>by </b><i><a class="txt" href="lattice5.html#E4:82_1_2"><i><font color="Green" title="E88">A33</font></i></a></i>;<br/><a NAME="E16:82_1_2"/><i><font color="Green" title="E93">A38</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">dq1</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="c14">Aq1</font>,<font color="Maroon" title="c14">Aq1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c27">z1'</font>
;<br/><b>take </b>
<font color="Maroon" title="c28">z2'</font>
;<br/><b>take </b>
<font color="Maroon" title="c29">z3'</font>
;<br/><b>thus </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c27">z1'</font> = 
<font color="Maroon" title="c15">dq1</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="c30">x''</font>,<font color="Maroon" title="c24">z1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E19:82"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="lattice5.html#E16:82_1_2"><i><font color="Green" title="E93">A38</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E8:82_1_2"><i><font color="Green" title="E89">A34</font></i></a></i>
;<br/><b>thus </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c28">z2'</font>,<font color="Maroon" title="c29">z3'</font> = 
<font color="Maroon" title="c15">dq1</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="c25">z2</font>,<font color="Maroon" title="c26">z3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E19:82"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="lattice5.html#E16:82_1_2"><i><font color="Green" title="E93">A38</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E9:82_1_2"><i><font color="Green" title="E90">A35</font></i></a></i>
;<br/><b>thus </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c27">z1'</font>,<font color="Maroon" title="c28">z2'</font> = 
<font color="Maroon" title="c15">dq1</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="c24">z1</font>,<font color="Maroon" title="c25">z2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E19:82"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="lattice5.html#E16:82_1_2"><i><font color="Green" title="E93">A38</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E10:82_1_2"><i><font color="Green" title="E91">A36</font></i></a></i>
;<br/><b>thus </b><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c29">z3'</font>,<font color="Maroon" title="c6">y</font> = 
<font color="Maroon" title="c15">dq1</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="c26">z3</font>,<font color="Maroon" title="c31">y''</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E19:82"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="lattice5.html#E16:82_1_2"><i><font color="Green" title="E93">A38</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="txt" href="lattice5.html#E11:82_1_2"><i><font color="Green" title="E92">A37</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
