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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a><br/> for <font color="Olive" title="b2">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b3">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds  <font color="Olive" title="b2">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds  <font color="Olive" title="b1">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">d</font> be   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> holds  <font color="Maroon" title="c3">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Olive" title="b1">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">g</font> =  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font>;<br/>
<a NAME="E1:23"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E2:23"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</font> holds <br/><font color="Maroon" title="c3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</font> implies <font color="Maroon" title="c3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:23_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x9</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">y9</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E3:23_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<font color="Maroon" title="c3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c9">x9</font>,<font color="Maroon" title="c10">y9</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x9</font>,<font color="Maroon" title="c10">y9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D5" target="_self" title="LATTICE8:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/>
<span class="kw">hence </span><a NAME="E6:23_1"/>
<font color="Maroon" title="c3">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:23"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:23"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:23"/>
<font color="Maroon" title="c3">d</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
