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<b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">h</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/><a NAME="E1:29_1"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="real_lat.html#E28"><i><font color="Green" title="E19">Lm8</font></i></a></i>;<br/><a NAME="E2:29_1"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="real_lat.html#E28"><i><font color="Green" title="E19">Lm8</font></i></a></i>;<br/><a NAME="E3:29_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="real_lat.html#E28"><i><font color="Green" title="E19">Lm8</font></i></a></i>;<br/><a NAME="E4:29_1"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="real_lat.html#E28"><i><font color="Green" title="E19">Lm8</font></i></a></i>;<br/><a NAME="E5:29_1"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="real_lat.html#E28"><i><font color="Green" title="E19">Lm8</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="real_lat.html#D5" target="_self" title="REAL_LAT:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="real_lat.html#D6" target="_self" title="REAL_LAT:def.6">Def6</a></i>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="real_lat.html#E5:29_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_lat.html#E3:29_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_lat.html#T14" target="_self" title="REAL_LAT:th.14">Th14</a></i>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_lat.html#E1:29_1"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="real_lat.html#E2:29_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_lat.html#K2" title="REAL_LAT:func.2">maxreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="real_lat.html#E4:29_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="real_lat.html#D5" target="_self" title="REAL_LAT:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_lat.html#K1" title="REAL_LAT:func.1">minreal</a>  <a href="funcsdom.html#K4" title="FUNCSDOM:func.4">.:</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="real_lat.html#D6" target="_self" title="REAL_LAT:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="real_lat.html#D6" target="_self" title="REAL_LAT:def.6">Def6</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:29"/>
<span class="p1">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="real_lat.html#K4" title="REAL_LAT:func.4">maxfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_lat.html#K5" title="REAL_LAT:func.5">minfuncreal</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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