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<b>let </b><font color="Maroon" title="c1">x0</font>, <font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  holds <br/>( <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> &amp; <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> )</span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  implies ( <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> &amp; <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> ) )</span><br/>







<b>assume </b><a NAME="E1:51"/>
<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> &amp; <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> )</span><br/>

<a NAME="E2:51"/><b>then </b>
( <font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x1</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font> &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></i>;<br/>
<a NAME="E3:51"/><b>then </b><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></i>;<br/>
<a NAME="E4:51"/><b>then </b>
<span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span>
 
<b>by </b><i><a class="ref" href="diff_1.html#T34" target="_self" title="DIFF_1:th.34">Th34</a></i>;<br/>
<b>hence </b><a NAME="E5:51"/>
( <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> &amp; <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="diff_1.html#K13" title="DIFF_1:func.13">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c3">x2</font></span><a href="diff_1.html#K13" title="DIFF_1:func.13">!]</a></span> )
 <b>by </b><i><a class="txt" href="diff_1.html#E3:51"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="diff_1.html#T34" target="_self" title="DIFF_1:th.34">Th34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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