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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="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">  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="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">g</font> &amp;  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> holds <br/><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 href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span><br/><span class="kw">let </span><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> ; <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> &amp;  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> implies <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 href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:185"/><span class="lab"><font color="Green" title="E86">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"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> or <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 href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:185"/>
 not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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"> <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 href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span><br/>

<a NAME="E3:185"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A2</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c4">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T107" title="FUNCT_4:th.107">FUNCT_4:107</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="185_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:185_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font> or  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="185_1_1"><span class="kw">suppose </span></a><a NAME="E1:185_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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"> <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 href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span><br/><div class="add"><a NAME="E2:185_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">g</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c4">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D3" target="_self" title="FUNCT_7:def.3">Def3</a></span>;<br/><a NAME="E3:185_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">g</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:185"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="funct_4.html#T123" title="FUNCT_4:th.123">FUNCT_4:123</a></span>;<br/><span class="kw">hence </span><a NAME="E4:185_1_1"/>
<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 href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:185"><span class="lab"><font color="Green" title="E87">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="185_1_2"><span class="kw">suppose </span></a><a NAME="E1:185_1_2"/>
 not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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"> <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 href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:185_1_2"/>
<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 href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:185"><span class="lab"><font color="Green" title="E86">A1</font></span></a>, <a class="ref" href="funct_7.html#D3" target="_self" title="FUNCT_7:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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