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<span class="kw">let </span><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font> be   <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> implies  <a href="valued_1.html#K64" title="VALUED_1:func.64">FirstLoc</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="valued_1.html#K64" title="VALUED_1:func.64">FirstLoc</a> <font color="Maroon" title="c1">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:215"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="valued_1.html#K64" title="VALUED_1:func.64">FirstLoc</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="valued_1.html#K64" title="VALUED_1:func.64">FirstLoc</a> <font color="Maroon" title="c1">F</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">N</font> =  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">M</font> =  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">F</font>;<br/>
<a NAME="E2:215"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:215"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="relat_1.html#T11" title="RELAT_1:th.11">RELAT_1:11</a></span>;<br/>
<a NAME="E3:215"/><span class="kw">then </span>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T60" title="XXREAL_2:th.60">XXREAL_2:60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:215"/>
 <a href="valued_1.html#K64" title="VALUED_1:func.64">FirstLoc</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="valued_1.html#K64" title="VALUED_1:func.64">FirstLoc</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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