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<span class="kw">let </span><font color="Maroon" title="c3">z1</font>, <font color="Maroon" title="c4">z2</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="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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="c3">z1</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> or  not <font color="Maroon" title="c4">z2</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> or  not <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z2</font> or <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:97_1_1"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<font color="Maroon" title="c3">z1</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E2:97_1_1"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
<font color="Maroon" title="c4">z2</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:97_1_1"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z2</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="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z2</font></span><br/>

<a NAME="E4:97_1_1"/><span class="lab"><font color="Green" title="E53">A4</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<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>,<span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D8" target="_self" onmouseover="rs('funct_3/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:97_1_1"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
<font color="Maroon" title="c5">x1</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="E7:97_1_1"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
<font color="Maroon" title="c6">y1</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="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E8:97_1_1"/><span class="lab"><font color="Green" title="E56">A7</font></span>: 
<font color="Maroon" title="c3">z1</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">x1</font>,<font color="Maroon" title="c6">y1</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="#E1:97_1_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T84" onmouseover="rs('zfmisc_1/T84')" onmouseout="rh()">ZFMISC_1:84</a></span>;<br/>
<a NAME="E9:97_1_1"/><span class="lab"><font color="Green" title="E57">A8</font></span>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:97_1_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E7:97_1_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="ref" href="funct_3.html#D8" target="_self" onmouseover="rs('funct_3/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">x2</font>, <font color="Maroon" title="c8">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E11:97_1_1"/><span class="lab"><font color="Green" title="E58">A9</font></span>: 
<font color="Maroon" title="c7">x2</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="E12:97_1_1"/><span class="lab"><font color="Green" title="E59">A10</font></span>: 
<font color="Maroon" title="c8">y2</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="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E13:97_1_1"/><span class="lab"><font color="Green" title="E60">A11</font></span>: 
<font color="Maroon" title="c4">z2</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="c7">x2</font>,<font color="Maroon" title="c8">y2</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="#E2:97_1_1"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E4:97_1_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T84" onmouseover="rs('zfmisc_1/T84')" onmouseout="rh()">ZFMISC_1:84</a></span>;<br/>
<a NAME="E14:97_1_1"/><span class="lab"><font color="Green" title="E61">A12</font></span>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">G</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">y2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:97_1_1"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="txt" href="#E12:97_1_1"><span class="lab"><font color="Green" title="E59">A10</font></span></a>, <a class="ref" href="funct_3.html#D8" target="_self" onmouseover="rs('funct_3/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E15:97_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</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="c7">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_1_1"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="txt" href="#E8:97_1_1"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="txt" href="#E13:97_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="txt" href="#E9:97_1_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E16:97_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A13</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:97_1_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E11:97_1_1"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E17:97_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_1_1"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="txt" href="#E8:97_1_1"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="txt" href="#E13:97_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="txt" href="#E9:97_1_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="txt" href="#E14:97_1_1"><span class="lab"><font color="Green" title="E61">A12</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:97_1_1"/>
<font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:97_1_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="txt" href="#E8:97_1_1"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="txt" href="#E12:97_1_1"><span class="lab"><font color="Green" title="E59">A10</font></span></a>, <a class="txt" href="#E13:97_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="txt" href="#E16:97_1_1"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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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