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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds   <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</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">  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</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="relat_1.html#D2" onmouseover="rs('relat_1/D2')" onmouseout="rh()">RELAT_1:def 2</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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  ) &amp; (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> ) )</span><br/>





<span class="kw">thus </span><a NAME="E1:75"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:75_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:75_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:75_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:75_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="funcop_1.html#D1" target="_self" onmouseover="rs('funcop_1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>

<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E4:75_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="funcop_1.html#D1" target="_self" onmouseover="rs('funcop_1/D1')" onmouseout="rh()">Def1</a></span>
;<br/>
<a NAME="E7:75_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:75_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</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">assume </span><a NAME="E2:75"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span></span><br/>

<a NAME="E3:75"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D7" onmouseover="rs('relat_1/D7')" onmouseout="rh()">RELAT_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:75"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E6:75"/>
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="funcop_1.html#D1" target="_self" onmouseover="rs('funcop_1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:75"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">~</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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>
