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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">E</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">F</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span><br/> for <font color="Olive" title="b3">TS</font> being   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Olive" title="b2">F</font>  st not  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Olive" title="b1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Olive" title="b3">TS</font></span>)</span> holds <br/> not <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">z</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">z</font>,<font color="Olive" title="b3">TS</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">E</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span><br/> for <font color="Olive" title="b2">TS</font> being   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Olive" title="b1">F</font>  st not  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c4">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Olive" title="b2">TS</font></span>)</span> holds <br/> not <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">z</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">z</font>,<font color="Olive" title="b2">TS</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">TS</font> being   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Maroon" title="c5">F</font>  st not  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c4">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Olive" title="b1">TS</font></span>)</span> holds <br/> not <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">z</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">z</font>,<font color="Olive" title="b1">TS</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">TS</font> be    <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Maroon" title="c5">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c4">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Maroon" title="c6">TS</font></span>)</span> implies  not <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">z</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c6">TS</font> )</span><br/>











<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
not  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c4">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Maroon" title="c6">TS</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">z</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c6">TS</font></span><br/>

<span class="kw">assume </span><a NAME="E2:33"/>
<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">z</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c6">TS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">w</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c4">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E4:33"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font>
 <span class="kw">and </span><br/><a NAME="E5:33"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c8">w</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">TS</font>
 <span class="kw">and </span><br/><a NAME="E6:33"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#D3" target="_self" title="REWRITE3:def.3">Def3</a></span>;<br/>
<a NAME="E7:33"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c8">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c3">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> the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Maroon" title="c6">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="rewrite3.html#D2" target="_self" title="REWRITE3:def.2">Def2</a></span>;<br/>
<a NAME="E8:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c8">w</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Maroon" title="c6">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T20" title="RELAT_1:th.20">RELAT_1:20</a></span>;<br/>
<a NAME="E9:33"/>
<font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c4">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="flang_2.html#T4" title="FLANG_2:th.4">FLANG_2:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:33"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="relat_1.html#T20" title="RELAT_1:th.20">RELAT_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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