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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/>



<span class="kw">thus </span><a NAME="E1:2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c3">C</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c3">C</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:2_1"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E6:2_1"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<a NAME="E7:2_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E8:2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E9:2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:2_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E9:2_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c3">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c3">C</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:2"/><span class="lab"><font color="Green" title="E4">A7</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">and </span><br/><a NAME="E5:2"/><span class="lab"><font color="Green" title="E5">A8</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E6:2"/><span class="lab"><font color="Green" title="E6">A9</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E7:2"/><span class="lab"><font color="Green" title="E7">A10</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2"><span class="lab"><font color="Green" title="E5">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E8:2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2"><span class="lab"><font color="Green" title="E5">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E9:2"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2"><span class="lab"><font color="Green" title="E4">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A11</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E11:2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2"><span class="lab"><font color="Green" title="E6">A9</font></span></a>, <a class="txt" href="#E7:2"><span class="lab"><font color="Green" title="E7">A10</font></span></a>, <a class="txt" href="#E10:2"><span class="lab"><font color="Green" title="E8">A11</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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