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<span class="kw">set </span><font color="Maroon" title="c5">h</font> =  <a href="valued_2.html#K16" title="VALUED_2:func.16">&lt;-&gt;</a> <font color="Maroon" title="c4">f</font>;<br/>
<a NAME="E1:129_1_1"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="valued_2.html#K16" title="VALUED_2:func.16">&lt;-&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_2.html#D33" target="_self" title="VALUED_2:def.33">Def33</a></span>;<br/>
<a NAME="E2:129_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="valued_2.html#K16" title="VALUED_2:func.16">&lt;-&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="valued_2.html#K6" title="VALUED_2:func.6">R_PFuncs</a> <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="valued_2.html#K16" title="VALUED_2:func.16">&lt;-&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="valued_2.html#K6" title="VALUED_2:func.6">R_PFuncs</a> <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:129_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="valued_2.html#K16" title="VALUED_2:func.16">&lt;-&gt;</a> <font color="Maroon" title="c4">f</font></span>)</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="valued_2.html#K6" title="VALUED_2:func.6">R_PFuncs</a> <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:129_1_1_1"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c7">x</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> <span class="p1">(<span class="default"><a href="valued_2.html#K16" title="VALUED_2:func.16">&lt;-&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:129_1_1_1"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<span class="p1">(<span class="default"><a href="valued_2.html#K16" title="VALUED_2:func.16">&lt;-&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129_1_1_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a></span>;<br/>
<a NAME="E6:129_1_1_1"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<span class="p1">(<span class="default"><a href="valued_2.html#K16" title="VALUED_2:func.16">&lt;-&gt;</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:129_1_1_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="valued_2.html#D33" target="_self" title="VALUED_2:def.33">Def33</a></span>;<br/>
<a NAME="E7:129_1_1_1"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:129_1_1_1_1"/>
(  not <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129_1_1_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E6:129_1_1_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</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="E8:129_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129_1_1"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E3:129_1_1_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></span>;<br/>
<a NAME="E9:129_1_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Y</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E10:129_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E11:129_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:129_1_1_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E6:129_1_1_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>;<br/>
<a NAME="E12:129_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:129_1_1_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E7:129_1_1_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:129_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="valued_2.html#K6" title="VALUED_2:func.6">R_PFuncs</a> <span class="p1">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_2.html#D12" target="_self" title="VALUED_2:def.12">Def12</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>
<span class="kw">hence </span><a NAME="E3:129_1_1"/>
 <a href="valued_2.html#K15" title="VALUED_2:func.15">&lt;-&gt;</a> <font color="Maroon" title="c4">f</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="valued_2.html#K6" title="VALUED_2:func.6">R_PFuncs</a> <span class="p2">(<span class="default"><a href="valued_2.html#K1" title="VALUED_2:func.1">DOMS</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:129_1_1"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</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>
