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<span class="kw">let </span><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">z</font>, <font color="Maroon" title="c5">y2</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 the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Olive" title="b3">TS</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp; <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b3">TS</font> &amp; <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b3">TS</font> holds <br/><font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">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="c6">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 the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Olive" title="b2">TS</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp; <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b2">TS</font> &amp; <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b2">TS</font> holds <br/><font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">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="c6">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="c7">F</font>  st the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Olive" title="b1">TS</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp; <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b1">TS</font> &amp; <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b1">TS</font> holds <br/><font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">TS</font> be    <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Maroon" title="c7">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Maroon" title="c8">TS</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp; <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c8">TS</font> &amp; <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c8">TS</font> implies <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font> )</span><br/>















<span class="kw">assume </span><a NAME="E1:32"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Maroon" title="c8">TS</font> is   <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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c8">TS</font> or  not <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c8">TS</font> or <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c8">TS</font>
 <span class="kw">and </span><br/><a NAME="E3:32"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c8">TS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">w1</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c6">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E5:32"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font>
 <span class="kw">and </span><br/><a NAME="E6:32"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c10">w1</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c8">TS</font>
 <span class="kw">and </span><br/><a NAME="E7:32"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c9">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="rewrite3.html#D3" target="_self" title="REWRITE3:def.3">Def3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">v2</font>, <font color="Maroon" title="c12">w2</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c6">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E9:32"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font>
 <span class="kw">and </span><br/><a NAME="E10:32"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c12">w2</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c8">TS</font>
 <span class="kw">and </span><br/><a NAME="E11:32"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c11">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="rewrite3.html#D3" target="_self" title="REWRITE3:def.3">Def3</a></span>;<br/>
<a NAME="E12:32"/>
<font color="Maroon" title="c10">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E7:32"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E11:32"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="afinsq_1.html#T31" title="AFINSQ_1:th.31">AFINSQ_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:32"/>
<font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="rewrite3.html#T17" target="_self" title="REWRITE3:th.17">Th17</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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