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<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> implies <font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E3">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="sprect_1.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>suppose </b></a><a NAME="E1:4_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c2">f</font> = <font color="Maroon" title="c1">f</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><a NAME="E3:4_1_1"/>
<font color="Maroon" title="c2">f</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/><b>hence </b><a NAME="E4:4_1_1"/>
<font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><a NAME="E1:4_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:4_1_2"/><i><font color="Green" title="E4">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E4:4_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="sprect_1.html#E3:4_1_2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="funcop_1.html#T15" title="FUNCOP_1:th.15">FUNCOP_1:15</a></i>;<br/><b>hence </b><a NAME="E5:4_1_2"/>
<font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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