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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c2">q</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> &amp; <font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">q</font> )
 ;<br/>

<a NAME="E2:14"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<b>assume </b><a NAME="E2:14_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<a NAME="E3:14_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1"/>
(  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">p</font> or  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">p</font> )
 ;<br/>

<a NAME="E2:14_1_1"/><b>then </b>
( <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:14_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="roughs_1.html#E2:14_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:14_1"/><b>then </b>
( <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="txt" href="roughs_1.html#E1:14"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:14_1"/>
contradiction
 <b>by </b><i><a class="txt" href="roughs_1.html#E1:14"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="roughs_1.html#E1:14_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="roughs_1.html#E2:14_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
<font color="Maroon" title="c1">p</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>
 <b>by </b><i><a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></i>;<br/>


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