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<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="lattice5.html#M2" target="_self">ExtensionSeq</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">FS</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">FD</font> be   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">FS</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">FS</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon">FD</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  )
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E3:82"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:82"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E31:82"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:82"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="txt" href="lattice5.html#E33:82"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">A1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A1</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq1</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq1</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E10:82"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Aq1</font>,<font color="Maroon">dq1</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A1</font>,<font color="Maroon">d1</font>
 <b>and </b><br/><a NAME="E11:82"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">d1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Aq1</font>,<font color="Maroon">dq1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="lattice5.html#E15:82"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E12:82"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E13:82"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E14:82"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Aq1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E15:82"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">dq1</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E16:82"/>
<font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E17:82"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">d1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E31:82"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E18:82"/>
<font color="Maroon">dq1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E40:82"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E19:82"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">dq1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E31:82"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E20:82"/>
<font color="Maroon">Aq1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E21:82"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">Aq1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E31:82"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E22:82"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E33:82"><i><font color="Green">E65</font></i></a>, <a class="txt" href="lattice5.html#E35:82"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:82"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E31:82"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E26:82"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A2</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d2</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A2</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq2</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq2</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E28:82"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">Aq2</font>,<font color="Maroon">dq2</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A2</font>,<font color="Maroon">d2</font>
 <b>and </b><br/><a NAME="E29:82"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A2</font>,<font color="Maroon">d2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Aq2</font>,<font color="Maroon">dq2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="lattice5.html#E15:82"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E30:82"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E31:82"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E32:82"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">Aq2</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E33:82"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">dq2</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E34:82"/>
<font color="Maroon">d2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E35:82"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">d2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E31:82"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E36:82"/>
<font color="Maroon">dq2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E37:82"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">dq2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E31:82"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E38:82"/>
<font color="Maroon">Aq2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E39:82"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">Aq2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E31:82"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E40:82"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#D1">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">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> or <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">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">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><a NAME="E2:82_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E70</font></i></a>, , <a class="txt" href="lattice5.html#E17:82"><i><font color="Green">E54</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A2</font> <b>by </b><i><a class="txt" href="lattice5.html#E9:82_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice5.html#E10:82_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="lattice5.html#E11:82_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="lattice5.html#E17:82_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E4:82_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 ;<br/><a NAME="E5:82_1_1"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A2</font>,<font color="Maroon">A2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">d2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d2</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>

;<br/><b>then consider </b><font color="Maroon">z1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Aq2</font>, <font color="Maroon">z2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Aq2</font>, <font color="Maroon">z3</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Aq2</font><b> such that </b><br/><a NAME="E8:82_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">dq2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E9:82_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">dq2</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z2</font>,<font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E10:82_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">dq2</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E11:82_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">dq2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">z3</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E32:82"><i><font color="Green">E64</font></i></a>, , <a class="txt" href="lattice5.html#E14:82"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E12:82_1_1"/>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aq2</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aq2</font> &amp; <font color="Maroon">z3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aq2</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">z1'</font> = <font color="Maroon">z1</font>, <font color="Maroon">z2'</font> = <font color="Maroon">z2</font>, <font color="Maroon">z3'</font> = <font color="Maroon">z3</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> <b>by </b><i/>;<br/><a NAME="E14:82_1_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E15:82_1_1"/><b>then </b>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Aq2</font>
 <b>by </b><i>, , <a class="txt" href="lattice5.html#E17:82"><i><font color="Green">E54</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">x''</font> = <font color="Maroon">x'</font>, <font color="Maroon">y''</font> = <font color="Maroon">y'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Aq2</font> <b>by </b><i/>;<br/><a NAME="E17:82_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">dq2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Aq2</font>,<font color="Maroon">Aq2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">z1'</font>
;<br/><b>take </b>
<font color="Maroon">z2'</font>
;<br/><b>take </b>
<font color="Maroon">z3'</font>
;<br/><b>thus </b><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z1'</font> = 
<font color="Maroon">dq2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x''</font>,<font color="Maroon">z1</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z2'</font>,<font color="Maroon">z3'</font> = 
<font color="Maroon">dq2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">z3</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z1'</font>,<font color="Maroon">z2'</font> = 
<font color="Maroon">dq2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z3'</font>,<font color="Maroon">y</font> = 
<font color="Maroon">dq2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z3</font>,<font color="Maroon">y''</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><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">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:82_1_2"/><b>then </b>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice5.html#E5:82_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="lattice5.html#E17:82"><i><font color="Green">E54</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A1</font> <b>by </b><i><a class="txt" href="lattice5.html#E33:82"><i><font color="Green">E65</font></i></a>, <a class="txt" href="lattice5.html#E35:82"><i><font color="Green">E66</font></i></a>, , <a class="txt" href="lattice5.html#E10:82_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E4:82_1_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 ;<br/><a NAME="E5:82_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<font color="Maroon">A1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">d1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">d1</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>

;<br/><b>then consider </b><font color="Maroon">z1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Aq1</font>, <font color="Maroon">z2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Aq1</font>, <font color="Maroon">z3</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Aq1</font><b> such that </b><br/><a NAME="E8:82_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">dq1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E9:82_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">dq1</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z2</font>,<font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E10:82_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">dq1</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E11:82_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">dq1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">z3</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E32:82"><i><font color="Green">E64</font></i></a>, , <a class="txt" href="lattice5.html#E14:82"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E12:82_1_2"/>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aq1</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aq1</font> &amp; <font color="Maroon">z3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Aq1</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">z1'</font> = <font color="Maroon">z1</font>, <font color="Maroon">z2'</font> = <font color="Maroon">z2</font>, <font color="Maroon">z3'</font> = <font color="Maroon">z3</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> <b>by </b><i/>;<br/><a NAME="E14:82_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E15:82_1_2"/><b>then </b>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Aq1</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E70</font></i></a>, , <a class="txt" href="lattice5.html#E17:82"><i><font color="Green">E54</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">x''</font> = <font color="Maroon">x'</font>, <font color="Maroon">y''</font> = <font color="Maroon">y'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Aq1</font> <b>by </b><i><a class="txt" href="lattice5.html#E11:82_1_2"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E17:82_1_2"/><i><font color="Green">E91</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">dq1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Aq1</font>,<font color="Maroon">Aq1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">z1'</font>
;<br/><b>take </b>
<font color="Maroon">z2'</font>
;<br/><b>take </b>
<font color="Maroon">z3'</font>
;<br/><b>thus </b><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z1'</font> = 
<font color="Maroon">dq1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x''</font>,<font color="Maroon">z1</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="lattice5.html#E17:82_1_2"><i><font color="Green">E91</font></i></a></i>
;<br/><b>thus </b><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z2'</font>,<font color="Maroon">z3'</font> = 
<font color="Maroon">dq1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">z3</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z1'</font>,<font color="Maroon">z2'</font> = 
<font color="Maroon">dq1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z3'</font>,<font color="Maroon">y</font> = 
<font color="Maroon">dq1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z3</font>,<font color="Maroon">y''</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
